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 | |
4489 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,15)(11,16,21,24)(12,17,22,20)(13,18,23,19)(25,26)(27,31)(28,30)(29,32)(33,35)(34,39)(36,37)(38,40);(1,3,5,8)(2,6,4,7)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,40,38,39);(10,15)(13,23)(16,24)(17,20)(26,28)(29,37)(31,33)(34,38);(9,11)(10,16)(12,18)(13,20)(14,21)(15,24)(17,23)(19,22)(25,27)(26,31)(28,33)(29,38)(30,35)(32,40)(34,37)(36,39);(9,12)(10,17)(11,19)(13,16)(14,22)(15,20)(18,21)(23,24)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,40,38,39);(1,4,5,2)(3,7,8,6)(9,13)(10,18)(11,20)(12,16)(14,23)(15,19)(17,21)(22,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,39)(33,40)(35,38)Polycyclic presentation:
2^9|a^2=gh,b^2=g,e^2=i,f^2=gi,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7998894);