Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220200 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1782 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)(7,18)(8,23,27,52)(9,28)(10,32)(11,16)(12,17)(13,21)(14,26,40,25)(15,29)(19,34,46,38)(20,47)(22,36)(24,53,56,39)(30,41)(31,58)(33,37)(35,49)(42,45,50,57)(43,48)(44,63,65,66)(51,55,67,54)(59,61)(60,68,62,64);(1,2,4,3)(5,7,10,21,12,22,11,33)(6,13,16,18,17,37,32,36)(8,19,26,57,27,46,25,45)(9,20,29,48,31,49,30,59)(14,38,52,42,40,34,23,50)(15,35,28,43,41,47,58,61)(24,44,55,68,56,65,54,64)(39,62,67,63,53,60,51,66);(5,8,10,25,12,27,11,26)(6,14,16,23,17,40,32,52)(7,19,21,45,22,46,33,57)(9,24,29,54,31,56,30,55)(13,34,18,42,37,38,36,50)(15,39,28,51,41,53,58,67)(20,44,48,64,49,65,59,68)(35,60,43,63,47,62,61,66);(1,3,4,2)(5,9,11,30,12,31,10,29)(6,15,32,58,17,41,16,28)(7,20,33,59,22,49,21,48)(8,24,26,55,27,56,25,54)(13,35,36,61,37,47,18,43)(14,39,52,67,40,53,23,51)(19,44,57,68,46,65,45,64)(34,60,50,66,38,62,42,63)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=fg,d^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276386);