Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454399 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60453 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1846 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)(8,14)(9,30)(10,17)(11,16)(12,18)(13,22)(15,33)(20,48)(21,36)(23,38)(24,37)(25,35)(26,56)(27,42)(28,41)(29,43)(31,60)(32,46)(34,45)(39,50)(40,57)(44,58)(47,55)(49,62)(51,63)(52,66)(53,59)(54,65)(61,68)(64,67);(5,7,10,22,12,23,11,24)(6,13,16,19,18,37,17,38)(8,25,27,50,29,51,28,20)(9,21,32,53,34,54,33,55)(14,39,41,35,43,48,42,63)(15,36,45,47,46,65,30,59)(26,52,31,61,58,67,57,49)(40,66,44,62,60,64,56,68);(5,8,12,29)(6,14,18,43)(7,20,23,50)(9,31,34,57)(10,28,11,27)(13,35,37,63)(15,44,46,56)(16,42,17,41)(19,39,38,48)(21,52,54,67)(22,51,24,25)(26,33,58,32)(30,60,45,40)(36,64,65,66)(47,62,59,68)(49,55,61,53);(2,4)(5,9,11,33,12,34,10,32)(6,15,17,30,18,46,16,45)(7,21,24,55,23,54,22,53)(8,26,28,57,29,58,27,31)(13,36,38,59,37,65,19,47)(14,40,42,56,43,60,41,44)(20,49,51,67,50,61,25,52)(35,62,39,66,63,68,48,64)Polycyclic presentation:
2^9|b^2=e,c^2=i,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361597);