#6393547

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×#3539	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3182 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)(9,10,14,20,15,19,13,18)(11,16,23,30,24,31,22,29)(12,17,25,32,26,33,27,34)(21,28,35,38,37,40,36,39);(1,3)(2,5)(4,8)(6,7)(9,11,15,24)(10,16,19,31)(12,21,26,37)(13,22,14,23)(17,28,33,40)(18,29,20,30)(25,35,27,36)(32,38,34,39);(9,12,14,27,15,26,13,25)(10,17,18,32,19,33,20,34)(11,21,23,36,24,37,22,35)(16,28,29,38,31,40,30,39);(3,8)(5,7)(9,13,15,14)(10,18,19,20)(11,22,24,23)(12,25,26,27)(16,29,31,30)(17,32,33,34)(21,35,37,36)(28,38,40,39);(2,6)(5,7)(9,13,15,14)(10,18,19,20)(11,23,24,22)(12,25,26,27)(16,30,31,29)(17,32,33,34)(21,36,37,35)(28,39,40,38)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=i,e^2=i,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, 6300743);