#431333

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

1082 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:

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

Polycyclic presentation:

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

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, 338529);