#443384

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fgi,c^2=fi,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 350580);