#7519174

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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

2666 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7426370);