#6686956

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682596
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	#30503	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1902 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6594152);