#453504

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

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×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1102 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)(7,10)(8,12)(9,11)(13,14,16,18)(15,17,19,20)(21,22,26,28,24,29,25,30)(23,27,33,35,31,34,32,36);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(15,19)(17,20)(21,23,24,31)(22,27,29,34)(25,32,26,33)(28,35,30,36);(6,11)(10,12)(13,15,16,19)(14,17,18,20)(23,31)(27,34)(32,33)(35,36);(2,4)(5,8,9,7)(6,12,11,10)(22,28)(25,26)(27,35)(29,30)(32,33)(34,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=gi,d^2=f,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, 360700);