Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394292 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,12,18)(7,19,24,36)(8,25,29,39)(9,30,32,42)(10,17,11,16)(13,23,37,22)(14,27,40,28)(15,33,41,31)(20,34,48,60)(21,49,51,35)(26,38,57,62)(43,46,53,47)(44,52,61,50)(45,63,66,68)(54,56,58,55)(59,65,67,64);(1,2)(3,4)(5,7,11,23,12,24,10,22)(6,13,17,36,18,37,16,19)(8,20,28,47,29,48,27,46)(9,21,33,52,32,51,31,50)(14,34,25,53,40,60,39,43)(15,35,42,61,41,49,30,44)(26,45,56,65,57,66,55,64)(38,59,54,68,62,67,58,63);(2,4)(5,8)(6,14)(7,20)(9,26)(10,27)(11,28)(12,29)(13,34)(15,38)(16,39)(17,25)(18,40)(19,43)(21,45)(22,46)(23,47)(24,48)(30,58)(31,55)(32,57)(33,56)(35,59)(36,53)(37,60)(41,62)(42,54)(44,63)(49,67)(50,64)(51,66)(52,65)(61,68);(5,9)(6,15)(7,21)(8,26)(10,31)(11,33)(12,32)(13,35)(14,38)(16,30)(17,42)(18,41)(19,44)(20,45)(22,50)(23,52)(24,51)(25,54)(27,55)(28,56)(29,57)(34,59)(36,61)(37,49)(39,58)(40,62)(43,63)(46,64)(47,65)(48,66)(53,68)(60,67)Polycyclic presentation:
2^9|a^2=i,b^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301490);