#353805

order	512
nilpotency class	3
isoclinism	isoclinic class of #17367
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)(11,19,21,15)(12,16)(13,18)(14,17)(20,31,33,26)(22,32,35,27)(23,28)(24,29)(25,30)(34,39,40,37)(36,38);(9,11,13,22,14,21,25,35)(10,15,18,27,17,19,30,32)(12,20,24,34,23,33,36,40)(16,26,29,37,28,31,38,39);(2,5)(6,8)(9,12)(10,16)(11,20)(13,24)(14,23)(15,26)(17,28)(18,29)(19,31)(21,33)(22,34)(25,36)(27,37)(30,38)(32,39)(35,40);(1,3,4,7)(2,6,5,8)(12,23)(16,28)(20,33)(24,36)(26,31)(29,38)(34,40)(37,39)

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,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, 261001);