#6584709

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1891 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)(7,11)(8,12)(9,10)(13,14,19,23)(15,20,26,37)(16,27,31,43)(17,22,18,33)(21,29,39,30)(24,40,28,42)(25,35,32,36)(34,44,38,41);(1,2)(3,4)(6,10)(11,12)(13,15)(14,20)(16,28)(17,32)(18,25)(19,26)(21,38)(22,36)(23,37)(24,31)(27,42)(29,41)(30,44)(33,35)(34,39)(40,43);(2,4)(5,7,9,8)(6,11,10,12)(13,16,18,30,19,31,17,29)(14,21,22,27,23,39,33,43)(15,24,25,41,26,28,32,44)(20,34,36,40,37,38,35,42);(1,2)(3,4)(5,8,9,7)(6,12,10,11);(2,4)(13,17,19,18)(14,22,23,33)(15,25,26,32)(16,29,31,30)(20,35,37,36)(21,27,39,43)(24,41,28,44)(34,42,38,40)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,e^d=ehi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491905);