Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1560 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)(9,10,12,14)(11,15,16,13)(17,18,21,25)(19,27,31,23)(20,24,32,38)(22,26,34,40)(28,37,30,35)(29,41,43,36)(33,39,45,48)(42,47,44,46);(1,3)(2,5)(4,8)(6,7)(17,19,21,31)(18,23,25,27)(20,29,32,43)(22,28,34,30)(24,36,38,41)(26,35,40,37)(33,42,45,44)(39,46,48,47);(9,11,12,16)(10,13,14,15)(19,28)(22,34)(23,35)(26,40)(27,37)(29,42)(30,31)(33,45)(36,46)(39,48)(41,47)(43,44);(2,6)(5,7)(17,20)(18,24)(19,29)(21,32)(22,33)(23,36)(25,38)(26,39)(27,41)(28,42)(30,44)(31,43)(34,45)(35,46)(37,47)(40,48);(3,8)(5,7)(19,30)(20,32)(22,34)(23,37)(24,38)(26,40)(27,35)(28,31)(29,42)(36,46)(41,47)(43,44)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6595288);