Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219706 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6,11,18,12,16,10,17)(7,19,23,37,24,13,22,38)(8,25,29,41,28,45,34,14)(9,30,31,15,32,44,33,43)(20,47,26,35,49,39,54,46)(21,50,51,42,52,61,53,36)(27,40,60,55,58,65,59,64)(48,66,56,67,68,62,57,63);(5,7,12,24)(6,13,16,19)(8,26,28,54)(9,21,32,52)(10,23,11,22)(14,39,41,47)(15,42,43,36)(17,38,18,37)(20,34,49,29)(25,35,45,46)(27,57,58,56)(30,61,44,50)(31,53,33,51)(40,63,65,67)(48,60,68,59)(55,62,64,66);(2,4)(5,8,11,34,12,28,10,29)(6,14,17,25,16,41,18,45)(7,20,23,54,24,49,22,26)(9,27,31,59,32,58,33,60)(13,35,37,39,19,46,38,47)(15,40,30,55,43,65,44,64)(21,48,51,57,52,68,53,56)(36,62,61,63,42,66,50,67);(2,4)(5,9,12,32)(6,15,16,43)(7,21,24,52)(8,27,28,58)(10,33,11,31)(13,36,19,42)(14,40,41,65)(17,30,18,44)(20,48,49,68)(22,53,23,51)(25,55,45,64)(26,56,54,57)(29,60,34,59)(35,62,46,66)(37,61,38,50)(39,63,47,67)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=eh,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,h^c=hi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295571);