#416103

order	512
nilpotency class	3
isoclinism	isoclinic class of #393963
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

1022 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)(9,10,13,14)(11,15,12,16)(17,18)(19,23)(20,25)(21,24)(22,26)(27,30)(28,32)(29,31);(1,3,5,4)(2,6,7,8)(10,14)(15,16)(17,19,22,28)(18,23,26,32)(20,27,21,29)(24,31,25,30);(1,4,5,3)(2,6,7,8)(9,11,13,12)(10,15,14,16)(18,24)(20,21)(23,30)(25,26)(27,29)(31,32);(9,12,13,11)(10,15,14,16)(17,20,22,21)(18,24,26,25)(19,27,28,29)(23,31,32,30)

Polycyclic presentation:

2^9|a^2=e,b^2=eh,c^2=e,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323299);