#7658728

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

4506 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)(11,18)(12,16)(13,15)(14,17)(19,24)(20,23)(21,22)(26,32)(28,37)(33,34)(35,39);(1,2,4,5)(3,6,7,8)(12,20)(16,23)(19,21)(22,24)(25,26,31,32)(27,34,30,33)(28,38,37,29)(35,36,39,40);(9,11,14,13)(10,15,17,18)(12,19,20,21)(16,22,23,24)(25,27,31,30)(26,33,32,34)(28,35,37,39)(29,36,38,40);(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,28)(26,29)(27,35)(30,39)(31,37)(32,38)(33,36)(34,40);(9,13,14,11)(10,15,17,18)(12,21,20,19)(16,22,23,24)(25,29)(26,28)(27,36)(30,40)(31,38)(32,37)(33,35)(34,39);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,22)(20,23)(21,24)(25,30,31,27)(26,33,32,34)(28,39,37,35)(29,36,38,40)

Polycyclic presentation:

2^9|b^2=g,c^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565924);