#386199

order	512
nilpotency class	3
isoclinism	isoclinic class of #220210
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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

1150 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)(3,5)(4,7)(6,8)(9,10)(11,16)(12,25,28,17)(13,29,21,18)(14,20)(15,19)(22,40,39,32)(23,34)(24,33)(26,37,30,35)(27,36)(31,38);(1,3)(2,4)(5,8)(6,7)(9,11,15,24)(10,16,19,33)(12,26,28,30)(13,22,21,39)(14,27,31,23)(17,35,25,37)(18,32,29,40)(20,36,38,34);(1,4,6,5)(2,7,8,3)(9,12,14,22,15,28,31,39)(10,17,20,32,19,25,38,40)(11,21,23,26,24,13,27,30)(16,29,34,35,33,18,36,37);(1,5,6,4)(2,7,8,3)(9,13)(10,18)(11,22)(12,27)(14,30)(15,21)(16,32)(17,36)(19,29)(20,37)(23,28)(24,39)(25,34)(26,31)(33,40)(35,38)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293395);