#358898

order	512
nilpotency class	3
isoclinism	isoclinic class of #188759
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1184 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266094);