#360858

order	512
nilpotency class	3
isoclinism	isoclinic class of #222256
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,8)(5,7)(9,10,16,22,14,20,15,21)(11,23,28,34,26,17,27,35)(12,19,30,38,13,18,31,37)(24,32,29,36,40,39,25,33);(1,3,5,4)(2,6,7,8)(9,11,14,26)(10,17,20,23)(12,29,13,25)(15,28,16,27)(18,36,19,33)(21,35,22,34)(24,31,40,30)(32,38,39,37);(1,4,5,3)(2,6,7,8)(9,12,14,13)(10,18,20,19)(11,24,26,40)(15,31,16,30)(17,32,23,39)(21,38,22,37)(25,28,29,27)(33,35,36,34);(9,13,14,12)(10,19,20,18)(11,25,26,29)(15,31,16,30)(17,33,23,36)(21,38,22,37)(24,28,40,27)(32,35,39,34)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,h^b=hi,g^c=gi,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, 268054);