#7614029

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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

2703 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)(20,25)(23,37)(24,32)(26,33)(27,31)(28,36)(29,35)(34,38)(39,40);(2,4)(6,7)(9,11,13,24,16,25,15,14)(10,17,30,20,21,32,19,22)(12,23,29,39,28,38,26,27)(18,31,33,34,36,40,35,37);(5,6,8,7)(9,12)(10,18)(11,23)(13,29)(14,27)(15,26)(16,28)(17,31)(19,35)(20,34)(21,36)(22,37)(24,39)(25,38)(30,33)(32,40);(1,2,3,4)(5,7,8,6)(9,13,16,15)(10,19,21,30)(11,24,25,14)(12,26,28,29)(17,22,32,20)(18,33,36,35)(23,27,38,39)(31,34,40,37);(9,14,15,25,16,24,13,11)(10,20,19,17,21,22,30,32)(12,27,26,38,28,39,29,23)(18,34,35,31,36,37,33,40)

Polycyclic presentation:

2^9|b^2=fi,c^2=g,d^2=hi,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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7521225);