#6576313

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1988 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:

(5,6,8,9)(7,10,11,12)(13,14,18,23)(15,24,26,20)(16,29,31,36)(17,22,19,32)(21,33,37,30)(25,40,41,44)(27,39,28,35)(34,42,38,43);(1,2)(3,4)(6,9)(10,12)(13,15,18,26)(14,20,23,24)(16,25,31,41)(17,27,19,28)(21,34,37,38)(22,35,32,39)(29,44,36,40)(30,42,33,43);(2,4)(13,16,18,31)(14,21,23,37)(15,25,26,41)(17,30,19,33)(20,34,24,38)(22,36,32,29)(27,42,28,43)(35,40,39,44);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(15,26)(20,24)(25,41)(27,28)(34,38)(35,39)(40,44)(42,43);(2,4)(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=f,b^2=i,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483509);