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 | |
1260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,29,33)(27,35,38,31)(28,32,40,48)(30,34,42,50)(36,53,54,45)(37,51,43,46)(39,56,55,47)(41,49,44,52);(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27,29,38)(26,31,33,35)(28,39,40,55)(30,43,42,37)(32,47,48,56)(34,51,50,46)(36,44,54,41)(45,52,53,49);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28)(26,32)(27,36)(29,40)(30,44)(31,45)(33,48)(34,52)(35,53)(37,55)(38,54)(39,43)(41,42)(46,56)(47,51)(49,50);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(26,33)(28,40)(31,35)(34,50)(36,54)(39,55)(41,44)(46,51);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(27,37)(28,40)(30,42)(31,46)(32,48)(34,50)(35,51)(36,39)(38,43)(45,47)(53,56)(54,55)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fi,d^2=fgi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6616639);