Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394292 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| 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 | |
1554 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)(8,25,28,14)(9,30,32,42)(10,17,11,16)(13,23,37,22)(15,33,41,31)(20,45,48,43)(21,50,52,35)(26,56,58,60)(27,39,29,40)(34,47,54,49)(38,57,55,59)(44,53,62,51)(46,61,66,68)(63,65,64,67);(1,2)(3,4)(5,7,11,23,12,24,10,22)(6,13,17,36,18,37,16,19)(8,20,29,49,28,48,27,47)(9,21,33,53,32,52,31,51)(14,34,40,45,25,54,39,43)(15,35,42,62,41,50,30,44)(26,46,59,67,58,66,57,65)(38,61,56,64,55,68,60,63);(2,4)(5,8,10,27,12,28,11,29)(6,14,16,39,18,25,17,40)(7,20,22,47,24,48,23,49)(9,26,31,57,32,58,33,59)(13,34,19,43,37,54,36,45)(15,38,30,60,41,55,42,56)(21,46,51,65,52,66,53,67)(35,61,44,63,50,68,62,64);(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)(20,46)(22,51)(23,53)(24,52)(25,55)(27,57)(28,58)(29,59)(34,61)(36,62)(37,50)(39,60)(40,56)(43,63)(45,64)(47,65)(48,66)(49,67)(54,68)Polycyclic presentation:
2^9|a^2=i,b^2=ei,c^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 302005);