#7563156

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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

2824 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,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,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7470352);