#450305

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

966 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:

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

Polycyclic presentation:

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

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, 357501);