#433405

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1162 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)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,21,26,23,28)(19,29,30,25,20,24,31,32);(5,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19,22,30,21,20,23,31)(18,24,27,32,26,29,28,25);(5,6,8,7)(10,14)(13,16)(17,20,21,19)(18,25,26,32)(22,30,23,31)(24,27,29,28);(2,4)(5,7,8,6)(18,26)(24,29)(25,32)(27,28)

Polycyclic presentation:

2^9|a^2=eg,b^2=efi,c^2=e,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=dei,f^c=fi,g^c=gi

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, 340601);