#6873958

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	#38049	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1523 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)(3,7)(4,6)(5,8)(17,18,24,29,23,20,22,26)(19,25,30,32,21,28,27,31);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,23,21)(18,25,20,28)(22,27,24,30)(26,31,29,32);(3,8)(5,7)(10,14)(15,16)(17,20,23,18)(19,25,21,28)(22,26,24,29)(27,32,30,31);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,23,19)(18,25,20,28)(22,30,24,27)(26,31,29,32);(9,12,13,11)(10,15,14,16)(19,21)(25,28)(27,30)(31,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

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, 6781154);