#6576923

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×#3013	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2680 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)(7,12)(8,10)(9,11)(13,14)(15,20)(16,27)(17,22)(18,31)(19,23)(21,28)(24,39)(25,34)(26,36)(29,37)(30,35)(32,38)(33,41)(40,43)(42,44);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,19,26)(14,20,23,36)(16,24,29,42)(17,30,18,25)(21,33,38,43)(22,35,31,34)(27,39,37,44)(28,41,32,40);(2,4)(13,16)(14,21)(15,24)(17,28)(18,32)(19,29)(20,33)(22,37)(23,38)(25,40)(26,42)(27,31)(30,41)(34,39)(35,44)(36,43);(1,2)(3,4)(5,8,9,7)(6,10,11,12);(2,4)(13,17,19,18)(14,22,23,31)(15,25,26,30)(16,28,29,32)(20,34,36,35)(21,37,38,27)(24,40,42,41)(33,39,43,44)

Polycyclic presentation:

2^9|b^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,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, 6484119);