#7651973

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

4403 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,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, 7559169);