#416804

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1090 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)(5,6,8,10)(7,9,11,12)(13,14)(15,19)(16,17)(18,20)(22,26)(24,31)(27,32)(28,33)(30,35)(34,36);(5,7)(6,9)(8,11)(10,12)(14,17)(18,19)(21,22,24,26,25,28,31,33)(23,27,30,32,29,34,35,36);(2,4)(7,11)(9,12)(21,23)(22,27)(24,30)(25,29)(26,32)(28,34)(31,35)(33,36);(13,15,16,20)(14,18,17,19)(23,29)(27,34)(30,35)(32,36)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^c=di

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, 324000);