#433220

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

1078 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,9,8)(7,11,12,10)(13,14)(15,17)(16,18)(19,20)(21,22,25,29,26,30,24,28)(23,27,34,36,32,35,33,31);(5,7,9,12)(6,10,8,11)(13,15,16,19)(14,17,18,20)(21,23,25,34,26,32,24,33)(22,27,29,36,30,35,28,31);(14,18)(17,20)(21,22,26,30)(23,31,32,36)(24,29,25,28)(27,34,35,33);(2,4)(5,8,9,6)(7,10,12,11)(23,32)(27,35)(31,36)(33,34)

Polycyclic presentation:

2^9|a^2=eg,b^2=efi,c^2=i,d^2=e,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, 340416);