#6393372

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

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×C₂²⋊C₄
prime radical	#62759	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,d^b=df,e^b=ei,g^c=gi,h^c=hi,e^d=ei

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, 6300568);