#6393854

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	C₂⁴	C₂⁵
prime radical	C₂×#3559	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#125 ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3111 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=i,e^2=f,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, 6301050);