#6624321

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×#468
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

1330 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6531517);