#6544590

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533105
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1566 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=i,d^2=fgh,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6451786);