#6624303

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574031
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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1438 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6531499);