#6612930

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1628 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)(17,18,22,27)(19,24,20,25)(21,30,31,26)(23,29,32,28);(1,3,5,4)(2,6,7,8)(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,22)(21,29)(24,30)(25,26)(27,32)(28,31);(2,7)(6,8)(10,13)(11,15)(17,19,22,20)(18,24,27,25)(21,28,31,29)(23,26,32,30);(1,4,5,3)(2,8,7,6)(9,11)(10,14)(12,15)(13,16)(17,20,22,19)(18,25,27,24)(21,29,31,28)(23,30,32,26);(10,13)(11,15)(19,20)(23,32)(24,25)(28,29)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=i,d^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,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, 6520126);