Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234485 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(17,18,23,29,24,30,22,28)(19,25,34,42,35,43,33,41)(20,27,36,44,21,37,38,26)(31,40,45,48,32,46,47,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,24,35)(18,25,30,43)(20,31,21,32)(22,33,23,34)(26,39,44,48)(27,40,37,46)(28,41,29,42)(36,45,38,47);(10,13)(11,15)(17,20,24,21)(18,26,30,44)(19,31,35,32)(22,36,23,38)(25,39,43,48)(27,29,37,28)(33,45,34,47)(40,42,46,41);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,21,24,20)(18,27,30,37)(19,32,35,31)(22,36,23,38)(25,40,43,46)(26,29,44,28)(33,45,34,47)(39,42,48,41);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,22,24,23)(18,28,30,29)(19,33,35,34)(20,36,21,38)(25,41,43,42)(26,27,44,37)(31,45,32,47)(39,40,48,46)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7190879);