#6392726

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂³×C₄⋊C₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1899 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=dfi,e^b=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6299922);