Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | #33579 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,4)(5,6,11,16,12,18,34,17)(7,13,22,38,23,40,56,39)(8,24,26,42,28,58,27,60)(9,15,30,45,31,46,10,32)(14,41,43,59,44,33,47,25)(19,48,51,66,53,68,52,62)(20,36,21,54,29,37,57,65)(35,61,63,49,64,55,67,50);(1,2,3,4)(5,7)(6,13)(8,19)(9,29)(10,21)(11,22)(12,23)(14,35)(15,37)(16,38)(17,39)(18,40)(20,31)(24,48)(25,50)(26,51)(27,52)(28,53)(30,57)(32,54)(33,55)(34,56)(36,46)(41,61)(42,66)(43,63)(44,64)(45,65)(47,67)(49,59)(58,68)(60,62);(5,8)(6,14)(7,19)(9,25)(10,33)(11,26)(12,28)(13,35)(15,42)(16,43)(17,47)(18,44)(20,49)(21,55)(22,51)(23,53)(24,32)(27,34)(29,50)(30,41)(31,59)(36,62)(37,66)(38,63)(39,67)(40,64)(45,58)(46,60)(48,54)(52,56)(57,61)(65,68);(5,9,11,30,12,31,34,10)(6,15,16,45,18,46,17,32)(7,20,22,21,23,29,56,57)(8,25,26,41,28,59,27,33)(13,36,38,54,40,37,39,65)(14,42,43,58,44,60,47,24)(19,49,51,55,53,50,52,61)(35,62,63,48,64,66,67,68)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=g,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^c=fh,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403024);