#7519010

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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2672 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:

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

Polycyclic presentation:

2^9|b^2=fi,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^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, 7426206);