#6761274

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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

1365 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)(17,18,23,32,22,26,21,27)(19,20,28,30,25,24,29,31);(1,3,6,2)(4,8,5,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,25)(18,24,26,20)(21,29,23,28)(27,30,32,31);(9,11,12,16)(10,13,14,15)(17,20,22,24)(18,25,26,19)(21,30,23,31)(27,29,32,28);(1,4,6,5)(2,7,3,8)(18,26)(19,25)(27,32)(28,29);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(17,21,22,23)(18,27,26,32)(19,28,25,29)(20,30,24,31)

Polycyclic presentation:

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

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, 6668470);