#6678403

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#24439	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1542 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

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, 6585599);