#6554690

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,3)(5,6,10,9)(7,12,8,11)(13,14,17,32,18,23,19,22)(15,24,27,35,26,20,28,39)(16,29,33,21,31,36,30,37)(25,40,43,38,41,44,42,34);(1,2,4,3)(5,7,10,8)(6,11,9,12)(13,15)(14,20)(16,25)(17,27)(18,26)(19,28)(21,34)(22,35)(23,24)(29,44)(30,42)(31,41)(32,39)(33,43)(36,40)(37,38);(1,3,4,2)(5,6,10,9)(7,12,8,11)(13,16)(14,21)(15,25)(17,30)(18,31)(19,33)(20,34)(22,36)(23,37)(24,38)(26,41)(27,42)(28,43)(29,32)(35,40)(39,44);(5,8,10,7)(6,11,9,12)(15,26)(20,24)(25,41)(27,28)(34,38)(35,39)(40,44)(42,43);(5,9,10,6)(7,12,8,11)(13,17,18,19)(14,22,23,32)(15,27,26,28)(16,30,31,33)(20,35,24,39)(21,36,37,29)(25,42,41,43)(34,40,38,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fi,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6461886);