#6370841

order	512
nilpotency class	3
isoclinism	isoclinic class of #60372
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278037);