#72250806

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24033
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂³×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20438 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2088 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,4,3)(5,6,9,15,10,16,23,30,12,17,28,33,11,7,18,13)(8,20,24,35,25,32,36,21,26,19,34,29,27,14,22,31);(1,3,4,2)(5,7,10,17,12,16,11,6)(8,21,27,29,26,31,25,35)(9,15,23,13,28,33,18,30)(14,22,20,34,32,36,19,24);(6,13)(7,15)(8,22)(9,18)(10,11)(14,19)(16,33)(17,30)(20,32)(21,31)(23,28)(24,27)(25,34)(26,36);(5,8,23,36,11,27,9,24,12,26,18,22,10,25,28,34)(6,14,13,29,7,19,33,21,17,32,30,35,16,20,15,31)

Polycyclic presentation:

2^10|a^2=e,b^2=fg,d^2=efhi,e^2=i,f^2=j,h^2=j,i^2=j,b^a=befj,c^a=cefgh,c^b=cefghj,d^a=dh,d^b=defghi,d^c=defghi,e^b=ei,e^c=eij,f^b=fj,f^c=fj,g^a=gj,g^b=gj,g^d=gj,h^a=hj,h^c=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250849, #72252698, #72252755
maximal normal subgroup of	#72250849, #72252698, #72252755

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.6*F.7);;
gap> SetPower(coll, 4, F.5*F.6*F.8*F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.6*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.6*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);