#6379264

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342728
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×#525	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3284 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)(6,8)(9,10,14,20,15,19,13,18)(11,16,22,30,23,31,27,33)(12,24,25,34,26,17,28,32)(21,35,37,39,38,29,36,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,23)(10,16,19,31)(12,21,26,38)(13,27,14,22)(17,29,24,35)(18,33,20,30)(25,37,28,36)(32,40,34,39);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,21)(13,25)(14,28)(15,26)(16,29)(18,32)(19,24)(20,34)(22,36)(23,38)(27,37)(30,39)(31,35)(33,40);(2,5)(6,8);(9,13,15,14)(10,18,19,20)(11,22,23,27)(12,25,26,28)(16,30,31,33)(17,32,24,34)(21,36,38,37)(29,39,35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6286460);