#6566013

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1400 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=i,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,g^c=gi,e^d=eh

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, 6473209);