Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454372 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2675 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2017 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1803 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)(7,19)(8,25)(9,31)(10,17)(11,16)(12,18)(13,22)(14,29)(15,35)(20,48)(21,52)(23,39)(24,38)(26,36)(27,57)(28,42)(30,41)(32,62)(33,44)(34,45)(37,55)(40,59)(43,60)(46,51)(47,56)(49,68)(50,65)(53,66)(54,61)(58,64)(63,67);(5,7,10,22,12,24,11,23)(6,13,16,19,18,39,17,38)(8,26,28,51,30,20,29,50)(9,21,33,54,35,56,34,55)(14,36,41,65,42,48,25,46)(15,37,44,47,31,61,45,52)(27,58,32,63,60,49,59,53)(40,64,43,66,62,68,57,67);(2,4)(5,8)(6,14)(7,20)(9,32)(10,29)(11,28)(12,30)(13,36)(15,43)(16,25)(17,41)(18,42)(19,46)(21,53)(22,51)(23,50)(24,26)(27,33)(31,57)(34,60)(35,59)(37,66)(38,65)(39,48)(40,44)(45,62)(47,64)(49,54)(52,68)(55,58)(56,63)(61,67);(5,9,10,33,12,35,11,34)(6,15,16,44,18,31,17,45)(7,21,22,54,24,56,23,55)(8,27,28,32,30,60,29,59)(13,37,19,47,39,61,38,52)(14,40,41,43,42,62,25,57)(20,49,50,53,26,58,51,63)(36,64,65,66,48,68,46,67)Polycyclic presentation:
2^9|b^2=e,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361568);