#6621128

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×#447
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

1583 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6528324);