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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4585 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)(11,15)(12,22)(13,18)(14,16)(17,19)(20,21)(23,24)(25,26)(27,29)(28,32)(30,33)(31,34)(35,37)(36,38)(39,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,20)(13,23)(14,19)(16,17)(18,24)(21,22)(25,27,31,32)(26,28,34,29)(30,38,40,35)(33,37,39,36);(1,3,5,4)(2,7,6,8)(10,16)(11,19)(12,18)(21,23)(25,28,31,29)(26,32,34,27)(30,36,40,37)(33,38,39,35);(1,4,5,3)(2,8,6,7)(9,11)(10,17)(12,20)(13,21)(14,19)(15,16)(18,24)(22,23)(25,29,31,28)(26,32,34,27)(30,37,40,36)(33,38,39,35);(9,12)(10,13)(11,20)(14,18)(15,23)(16,22)(17,21)(19,24)(25,26)(27,28)(29,32)(30,39)(31,34)(33,40)(35,37)(36,38);(9,13)(10,18)(11,21)(12,16)(14,22)(15,20)(17,24)(19,23)(25,30)(26,33)(27,35)(28,36)(29,37)(31,40)(32,38)(34,39)Polycyclic presentation:
2^9|b^2=hi,c^2=g,d^2=g,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, 7876637);