#7519290

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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

2622 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)(9,10,14,20)(11,23,26,17)(12,29,25,39)(13,33,24,37)(15,22,16,21)(18,32,38,31)(19,34,40,30)(27,35,28,36);(1,2,4,3)(5,6)(7,8)(9,11)(10,17)(12,30)(13,31)(14,26)(15,28)(16,27)(18,37)(19,39)(20,23)(21,36)(22,35)(24,32)(25,34)(29,40)(33,38);(9,12,14,25)(10,18,20,38)(11,24,26,13)(15,32,16,31)(17,19,23,40)(21,39,22,29)(27,30,28,34)(33,35,37,36);(6,8)(9,13,14,24)(10,19,20,40)(11,25,26,12)(15,34,16,30)(17,18,23,38)(21,37,22,33)(27,31,28,32)(29,35,39,36);(1,3,4,2)(10,20)(11,26)(12,25)(19,40)(21,22)(27,28)(31,32)(33,37)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,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=cgi,d^b=dh,e^b=ei,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

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7426486);