#6370909

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342574
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

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	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4310 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)(3,6)(4,5)(7,8)(9,10,13,18,14,16,24,29)(11,15,20,27,21,25,33,36)(12,17,23,30,22,28,34,38)(19,26,32,37,31,35,39,40);(2,5)(6,8)(9,11,14,21)(10,15,16,25)(12,19,22,31)(13,20,24,33)(17,26,28,35)(18,27,29,36)(23,32,34,39)(30,37,38,40);(1,3,4,7)(2,6,5,8)(10,16)(13,24)(15,25)(17,28)(20,33)(23,34)(26,35)(32,39);(9,12)(10,17)(11,19)(13,23)(14,22)(15,26)(16,28)(18,30)(20,32)(21,31)(24,34)(25,35)(27,37)(29,38)(33,39)(36,40);(12,22)(17,28)(19,31)(23,34)(26,35)(30,38)(32,39)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278105);