Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #42967 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,19)(12,18)(13,16)(14,20)(15,17)(21,22)(23,24)(25,26)(27,30)(28,36)(29,31)(32,37)(33,35)(34,40)(38,39);(1,2)(3,7)(4,8)(5,6)(9,11,15,13)(10,16,17,19)(12,21,20,23)(14,22,18,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,40)(37,39);(1,3,5,4)(2,7,6,8)(10,17)(12,20)(16,19)(21,23)(26,29)(27,33)(28,37)(38,40);(9,12,15,20)(10,18,17,14)(11,21,13,23)(16,24,19,22)(25,28,31,37)(26,32,29,36)(27,34,33,39)(30,38,35,40);(9,13,15,11)(10,19,17,16)(12,23,20,21)(14,24,18,22)(25,29,31,26)(27,35,33,30)(28,36,37,32)(34,40,39,38);(1,4,5,3)(2,8,6,7)(9,14,15,18)(10,20,17,12)(11,22,13,24)(16,23,19,21)(25,30,31,35)(26,27,29,33)(28,38,37,40)(32,34,36,39)Polycyclic presentation:
2^9|b^2=hi,c^2=ghi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7893548);