#403720

order	512
nilpotency class	3
isoclinism	isoclinic class of #393893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=egi,c^2=g,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310916);