#7563178

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2742 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7470374);