#382413

order	512
nilpotency class	3
isoclinism	isoclinic class of #29610
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#379
prime radical	C₂×#3449	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3576 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)(11,16)(12,24)(13,20)(14,18)(15,19)(17,25)(21,36)(22,35)(23,31)(26,33)(27,34)(28,30)(29,37)(32,39)(38,40);(2,5)(6,8)(9,11)(10,16)(12,21)(13,28)(14,22)(15,23)(17,32)(18,35)(19,31)(20,30)(24,36)(25,39)(26,37)(27,38)(29,33)(34,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,26,15,27,13,25)(10,17,20,34,19,33,18,24)(11,21,22,37,23,38,28,39)(16,29,30,36,31,32,35,40);(3,7)(6,8)

Polycyclic presentation:

2^9|c^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=de

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289609);