#6552602

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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1796 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)(13,14,17,23,18,22,32,21)(15,24,27,35,26,19,33,36)(16,28,31,37,30,38,29,20)(25,40,43,34,41,44,42,39);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,25)(17,27)(18,26)(20,34)(21,35)(22,24)(23,36)(28,44)(29,42)(30,41)(31,43)(32,33)(37,39)(38,40);(1,3,4,2)(6,9)(10,12)(13,16,18,30)(14,20,22,37)(15,25,26,41)(17,29,32,31)(19,34,24,39)(21,28,23,38)(27,42,33,43)(35,44,36,40);(5,7)(6,10)(8,11)(9,12)(15,26)(19,24)(25,41)(27,33)(34,39)(35,36)(40,44)(42,43);(7,11)(10,12)(13,17,18,32)(14,21,22,23)(15,27,26,33)(16,29,30,31)(19,35,24,36)(20,28,37,38)(25,42,41,43)(34,44,39,40)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=f,e^2=i,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, 6459798);