#6756160

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1384 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

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, 6663356);