#7614075

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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2661 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=g,d^2=h,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, 7521271);