#7563202

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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

2769 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)(9,10,15,18)(11,21,13,29)(12,25,27,17)(14,31,23,20)(16,30,19,24)(22,36,26,35)(28,34,37,39)(32,38,33,40);(1,2)(3,4)(6,7)(9,11,14,24,15,13,23,30)(10,16,20,21,18,19,31,29)(12,26,28,40,27,22,37,38)(17,33,34,36,25,32,39,35);(5,6,8,7)(9,12,15,27)(10,17,18,25)(11,22,13,26)(14,28,23,37)(16,32,19,33)(20,34,31,39)(21,35,29,36)(24,38,30,40);(2,4)(5,7,8,6)(10,18)(12,27)(16,19)(20,31)(21,29)(22,26)(28,37)(38,40);(9,13,14,30,15,11,23,24)(10,19,20,29,18,16,31,21)(12,22,28,38,27,26,37,40)(17,32,34,35,25,33,39,36)

Polycyclic presentation:

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

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, 7470398);