#6549036

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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×#493	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=i,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,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, 6456232);