#72250786

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24196
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂²×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20438 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2139 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=e,c^2=g,d^2=efgj,e^2=i,f^2=j,h^2=j,i^2=j,b^a=bghj,c^b=cefj,d^a=dh,d^b=di,d^c=defi,e^a=ej,e^c=ei,e^d=ej,f^c=fj,g^b=gj,g^d=gj,h^b=hj,h^c=hj,i^c=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250849, #72251004, #72252718
maximal normal subgroup of	#72250849, #72251004, #72252718

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