#7563286

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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

2793 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)(11,18,22,36)(12,17,24,32)(13,27,23,16)(14,30,29,19)(21,35,37,33)(25,31,28,40)(26,39,38,34);(1,2)(3,4)(6,7)(9,11,14,23,15,22,29,13)(10,16,19,18,20,27,30,36)(12,21,26,25,24,37,38,28)(17,31,34,35,32,40,39,33);(5,6,8,7)(9,12,15,24)(10,17,20,32)(11,21,22,37)(13,28,23,25)(14,26,29,38)(16,31,27,40)(18,35,36,33)(19,34,30,39);(2,4)(5,7,8,6)(11,22)(12,24)(13,23)(16,27)(17,32)(18,36)(26,38)(34,39);(9,13,29,22,15,23,14,11)(10,18,30,16,20,36,19,27)(12,25,38,21,24,28,26,37)(17,33,39,40,32,35,34,31)

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,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7470482);