Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395246 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,3)(5,6)(7,19)(8,25)(9,31)(10,17)(11,16)(12,18)(13,22)(14,29)(15,33)(20,39)(21,36)(23,38)(24,37)(26,54)(27,56)(28,41)(30,42)(32,44)(34,43)(35,49)(40,57)(45,48)(46,51)(47,66)(50,60)(52,63)(53,59)(55,65)(58,61)(62,67)(64,68);(1,2,4,3)(5,7,10,22,12,24,11,23)(6,13,16,19,18,38,17,37)(8,26,28,48,29,49,30,20)(9,21,32,50,34,52,33,51)(14,39,41,35,25,45,42,54)(15,36,43,46,44,63,31,60)(27,55,57,67,58,68,59,47)(40,64,56,62,53,65,61,66);(1,3,4,2)(5,8,10,28,12,29,11,30)(6,14,16,41,18,25,17,42)(7,20,22,26,24,48,23,49)(9,27,32,57,34,58,33,59)(13,35,19,45,38,54,37,39)(15,40,43,56,44,53,31,61)(21,47,50,55,52,67,51,68)(36,62,46,65,63,66,60,64);(5,9,10,32,12,34,11,33)(6,15,16,43,18,44,17,31)(7,21,22,50,24,52,23,51)(8,27,28,57,29,58,30,59)(13,36,19,46,38,63,37,60)(14,40,41,56,25,53,42,61)(20,47,26,55,48,67,49,68)(35,62,45,65,54,66,39,64)Polycyclic presentation:
2^9|b^2=e,c^2=e,d^2=hi,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,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319370);