#6696414

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	#24424	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1703 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,d^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6603610);