#6621134

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573549
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1481 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6528330);