#72250173

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24089
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#3058	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20384 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2206 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=e,c^2=g,d^2=efghij,e^2=i,f^2=j,i^2=j,b^a=bf,c^b=cefghj,d^a=dhij,d^c=defghij,e^a=ej,e^c=ei,f^c=fj,f^d=fj,g^b=gj,g^d=gj,h^b=hj,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250233, #72252721, #72252751, …
maximal normal subgroup of	#72250233, #72252721, #72252751, …

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.7);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.5*F.6*F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
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*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);