Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #482 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2034 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(1,2)(3,4)(5,6)(7,20,25,49)(8,27)(9,17)(10,16)(11,36)(12,19)(13,22,40,23)(14,29)(15,33)(18,35)(21,52,55,58)(24,50,57,41)(26,47,37,39)(28,61,63,42)(30,45)(31,44)(32,43)(34,48)(38,62,67,56)(46,65)(51,53,68,54)(59,64)(60,66);(5,7)(6,13)(8,28)(9,24)(10,22)(11,37)(12,25)(14,42)(15,47)(16,39)(17,49)(18,50)(19,40)(20,48)(21,30)(23,33)(26,35)(27,58)(29,62)(31,54)(32,63)(34,57)(36,41)(38,44)(43,52)(45,51)(46,61)(53,66)(55,64)(56,65)(59,68)(60,67);(2,4)(5,8,9,29,11,31,33,64,12,32,34,65,35,66,10,30)(6,14,15,43,18,45,17,44,19,46,16,27,36,59,48,60)(7,21,22,53,26,56,57,63,25,55,23,54,37,62,24,28)(13,38,20,51,41,52,39,42,40,67,49,68,50,58,47,61)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=h,e^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fh,g^b=gi,h^b=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55748);