Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806671 |
| 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 |
| prime radical | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,4)(9,10)(11,15)(12,16)(13,14)(17,18,23,28)(19,30,32,24)(20,34,33,25)(21,29,35,26)(22,38,39,27)(31,40,45,44)(36,41,46,47)(37,42,48,43);(1,2)(3,7)(4,8)(5,6)(17,19,23,32)(18,24,28,30)(20,35,33,21)(22,36,39,46)(25,26,34,29)(27,41,38,47)(31,37,45,48)(40,43,44,42);(1,3,5,4)(2,7,6,8)(10,14)(15,16)(17,20,21,19,23,33,35,32)(18,25,29,24,28,34,26,30)(22,36,37,31,39,46,48,45)(27,41,43,40,38,47,42,44);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(17,21,23,35)(18,26,28,29)(19,20,32,33)(22,37,39,48)(24,34,30,25)(27,42,38,43)(31,36,45,46)(40,47,44,41);(2,6)(7,8)(9,12,13,11)(10,16,14,15)(17,22,35,48,23,39,21,37)(18,27,26,42,28,38,29,43)(19,31,20,36,32,45,33,46)(24,40,25,41,30,44,34,47)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fhi,d^2=g,e^2=fghi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6750972);