Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418535 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1177 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)(7,18,22,13)(8,23,27,41)(9,15,29,43)(10,17)(11,32)(12,16)(14,26,40,33)(19,47)(20,35)(21,37,30,36)(24,46)(25,56)(28,44,31,45)(34,49)(38,50)(39,58)(42,53)(48,64,67,63)(51,61)(52,62)(54,59)(55,66,68,60)(57,65);(5,7,11,21,12,22,10,30)(6,13,32,36,16,18,17,37)(8,24,26,50,27,49,33,19)(9,20,28,52,29,53,31,51)(14,38,23,34,40,47,41,46)(15,42,44,61,43,35,45,62)(25,55,58,67,59,68,57,48)(39,64,54,66,65,63,56,60);(2,4)(5,8,12,27)(6,14)(7,19,22,50)(9,25,29,59)(10,26,11,33)(13,34)(15,39)(16,40)(17,23)(18,46)(20,48,53,67)(21,49,30,24)(28,57,31,58)(32,41)(35,60)(36,38)(37,47)(42,66)(43,65)(44,56)(45,54)(51,55,52,68)(61,64)(62,63);(5,9)(6,15,16,43)(7,20)(8,25)(10,31)(11,28)(12,29)(13,35,18,42)(14,39,40,65)(17,45,32,44)(19,48)(21,52)(22,53)(23,54,41,56)(24,55)(26,58)(27,59)(30,51)(33,57)(34,60,46,66)(36,62,37,61)(38,63,47,64)(49,68)(50,67)Polycyclic presentation:
2^9|a^2=efg,b^2=eg,c^2=ei,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348422);