#7518969

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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

2634 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=i,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^c=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, 7426165);