#6761342

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	#37954	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1356 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)(3,6)(4,8)(5,7)(17,18,23,25,22,24,21,26)(19,27,32,31,28,20,29,30);(1,3,6,2)(4,8,5,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,20)(21,29)(22,28)(23,32)(24,27)(25,30)(26,31);(9,11,12,16)(10,13,14,15)(17,20,22,27)(18,19,24,28)(21,30,23,31)(25,29,26,32);(1,4,6,5)(2,7,3,8)(18,24)(19,28)(25,26)(29,32);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(17,21,22,23)(18,25,24,26)(19,29,28,32)(20,30,27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6668538);