#466390

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

1116 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)(5,6,8,9)(7,10,11,12)(13,14,17,19)(15,18,16,20)(21,22)(23,31)(24,29)(25,28)(26,30)(27,33)(32,35)(34,36);(6,9)(10,12)(13,15,17,16)(14,18,19,20)(21,23,24,32,25,33,26,34)(22,27,29,36,28,31,30,35);(5,7,8,11)(6,10,9,12)(14,19)(18,20)(22,28)(23,32)(24,26)(27,35)(31,36)(33,34);(2,4)(7,11)(10,12)(13,16,17,15)(14,20,19,18)(21,24,25,26)(22,29,28,30)(23,32,33,34)(27,36,31,35)

Polycyclic presentation:

2^9|a^2=ef,b^2=fg,c^2=ei,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,g^c=gi

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, 373586);