#417224

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, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1132 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=fg,d^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 324420);