#7645881

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4383 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7553077);