#410674

order	512
nilpotency class	3
isoclinism	isoclinic class of #395154
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=efi,b^2=h,c^2=fgi,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=di

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, 317870);