#6848330

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667652
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	#34935	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1938 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)(9,10,13,16)(11,14,12,15)(17,18)(19,23)(20,27)(21,29)(22,25)(24,26)(28,31)(30,32);(1,3)(2,5)(4,8)(6,7)(11,12)(14,15)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,32,27);(3,8)(5,7)(9,11,13,12)(10,14,16,15)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29);(2,6)(5,7)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,32,27);(9,12,13,11)(10,15,16,14)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6755526);