#6393866

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342866
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3562	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3070 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,g^c=gi,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, 6301062);