#353829

order	512
nilpotency class	3
isoclinism	isoclinic class of #17386
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1698 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=f,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261025);