#6682992

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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×#3060	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2018 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)(8,10)(9,12)(13,14)(15,24)(16,29)(17,23)(18,20)(19,33)(21,28)(22,31)(25,39)(26,41)(27,34)(30,37)(32,36)(35,43)(38,40)(42,44);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(14,20)(15,25)(22,29)(23,33)(26,40)(27,28)(36,37)(42,43);(2,4)(13,15,19,28,18,25,17,27)(14,21,23,24,20,34,33,39)(16,26,30,42,31,40,32,43)(22,35,37,38,29,44,36,41);(6,10)(9,11)(13,16,18,31)(14,22,20,29)(15,26,25,40)(17,32,19,30)(21,35,34,44)(23,37,33,36)(24,38,39,41)(27,43,28,42);(2,4)(7,12)(9,11)(13,17,18,19)(14,23,20,33)(15,27,25,28)(16,30,31,32)(21,24,34,39)(22,36,29,37)(26,42,40,43)(35,41,44,38)

Polycyclic presentation:

2^9|b^2=fi,c^2=g,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=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, 6590188);