#7519185

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

2632 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=i,e^2=f,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, 7426381);