#415464

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

1134 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)(13,14,16,18)(15,19,20,17)(22,26)(24,27)(28,32)(29,33)(31,34)(35,36);(5,7)(6,9)(8,11)(10,12)(21,22,27,33,25,29,24,26)(23,28,34,36,30,35,31,32);(2,4)(7,11)(9,12)(21,23)(22,28)(24,31)(25,30)(26,32)(27,34)(29,35)(33,36);(13,15,16,20)(14,17,18,19)(23,30)(28,35)(31,34)(32,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=ei,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, 322660);