Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | |
1465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,24)(20,25)(21,26)(22,28)(23,27)(29,40)(30,32)(31,44)(33,41)(34,42)(35,39)(36,43)(37,38)(45,47)(46,48);(1,3,4,8)(2,5,6,7)(17,19,23,33)(18,24,27,41)(20,34,36,45)(21,30,28,44)(22,31,26,32)(25,42,43,47)(29,39,46,37)(35,48,38,40);(10,14)(15,16)(17,20,23,36)(18,25,27,43)(19,29,33,46)(21,38,28,35)(22,39,26,37)(24,40,41,48)(30,47,44,42)(31,34,32,45);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,23,28)(18,26,27,22)(19,30,33,44)(20,35,36,38)(24,32,41,31)(25,39,43,37)(29,42,46,47)(34,48,45,40);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(18,27)(19,31)(20,36)(22,26)(24,30)(29,45)(32,33)(34,46)(35,38)(40,42)(41,44)(47,48)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=gi,e^2=g,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=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6637614);