Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1862 |
| 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 | |
678 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)(5,6,9,17,12,18,11,15)(7,19,21,13,23,39,35,40)(8,24,26,57,28,58,36,47)(10,30,29,48,34,45,33,16)(14,32,43,61,44,63,42,27)(20,50,51,41,53,66,52,68)(22,54,31,46,56,38,62,67)(25,49,59,37,60,64,55,65);(5,7,9,21,12,23,11,35)(6,13,15,19,18,40,17,39)(8,25,26,59,28,60,36,55)(10,31,29,56,34,62,33,22)(14,41,42,50,44,68,43,66)(16,46,45,54,48,67,30,38)(20,32,51,61,53,63,52,27)(24,37,47,49,58,65,57,64);(2,4)(5,8)(6,14)(7,20)(9,26)(10,32)(11,36)(12,28)(13,37)(15,42)(16,47)(17,43)(18,44)(19,49)(21,51)(22,55)(23,53)(24,30)(25,31)(27,33)(29,61)(34,63)(35,52)(38,66)(39,64)(40,65)(41,46)(45,58)(48,57)(50,54)(56,59)(60,62)(67,68)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55323);