#425926

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1304 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)(5,6,9,8)(7,11,12,10)(13,14,16,18)(15,19,20,17)(21,22)(23,31)(24,29)(25,27)(26,30)(28,33)(32,35)(34,36);(5,7)(6,10)(8,11)(9,12)(13,15,16,20)(14,17,18,19)(22,27)(23,32)(24,26)(28,35)(31,36)(33,34);(5,8,9,6)(7,11,12,10)(14,18)(17,19)(21,23,25,33)(22,28,27,31)(24,32,26,34)(29,36,30,35);(2,4)(7,12)(10,11)(15,20)(17,19)(21,24,25,26)(22,29,27,30)(23,32,33,34)(28,36,31,35)

Polycyclic presentation:

2^9|a^2=ei,b^2=fi,c^2=efi,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,g^c=gi

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, 333122);