#7614080

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=h,e^2=fi,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, 7521276);