Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1763 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,23,36,22,28,21,27)(19,24,37,47,31,41,30,40)(20,32,33,25,35,26,34,42)(29,43,44,38,46,39,45,48);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(17,19,22,31)(18,24,28,41)(20,29,35,46)(21,30,23,37)(25,38,42,48)(26,39,32,43)(27,40,36,47)(33,44,34,45);(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,29)(21,33)(22,35)(23,34)(24,38)(26,27)(28,42)(30,44)(31,46)(32,36)(37,45)(39,40)(41,48)(43,47);(2,6)(5,7)(10,13)(14,16)(17,20)(18,26)(19,29)(21,34)(22,35)(23,33)(24,39)(25,27)(28,32)(30,45)(31,46)(36,42)(37,44)(38,40)(41,43)(47,48);(3,8)(5,7)(17,21,22,23)(18,27,28,36)(19,30,31,37)(20,33,35,34)(24,40,41,47)(25,26,42,32)(29,44,46,45)(38,39,48,43)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6664409);