Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395246 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1798 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6,12,18)(7,13,22,36)(8,14,25,39)(9,27,30,41)(10,17,11,16)(15,28,42,29)(19,33,46,59)(20,48,50,63)(21,43,31,35)(23,51,47,37)(24,53,54,64)(26,44,32,40)(34,56,60,49)(38,57,62,55)(45,65,66,61)(52,68,67,58);(5,7)(6,13)(8,23)(9,20)(10,31)(11,21)(12,22)(14,37)(15,34)(16,43)(17,35)(18,36)(19,26)(24,52)(25,47)(27,48)(28,56)(29,49)(30,50)(32,46)(33,40)(38,61)(39,51)(41,63)(42,60)(44,59)(45,55)(53,58)(54,67)(57,66)(62,65)(64,68);(2,4)(5,8)(6,14)(7,19)(9,24)(10,32)(11,26)(12,25)(13,33)(15,38)(16,44)(17,40)(18,39)(20,45)(21,47)(22,46)(23,31)(27,53)(28,57)(29,55)(30,54)(34,58)(35,51)(36,59)(37,43)(41,64)(42,62)(48,65)(49,67)(50,66)(52,56)(60,68)(61,63);(5,9,10,28,12,30,11,29)(6,15,16,41,18,42,17,27)(7,20,31,56,22,50,21,49)(8,24,32,57,25,54,26,55)(13,34,43,63,36,60,35,48)(14,38,44,64,39,62,40,53)(19,45,23,52,46,66,47,67)(33,58,37,61,59,68,51,65)Polycyclic presentation:
2^9|a^2=i,d^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318938);