Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24987 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1202 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)(7,13,22,36,21,35,31,43)(8,23,27,39,28,14,26,40)(9,15,30,42,29,41,32,44)(19,45,24,53,47,33,52,37)(20,48,50,60,51,34,49,61)(25,54,57,64,56,38,58,63)(46,59,66,68,67,65,55,62);(2,4)(5,7,10,22,12,21,11,31)(6,13,18,43,16,35,17,36)(8,24,26,47,28,52,27,19)(9,20,30,49,29,51,32,50)(14,37,40,33,23,53,39,45)(15,34,44,61,41,48,42,60)(25,55,57,67,56,66,58,46)(38,62,64,59,54,68,63,65);(5,8,12,28)(6,14,16,23)(7,19,21,47)(9,25,29,56)(10,27,11,26)(13,33,35,45)(15,38,41,54)(17,40,18,39)(20,46,51,67)(22,52,31,24)(30,58,32,57)(34,59,48,65)(36,53,43,37)(42,64,44,63)(49,66,50,55)(60,68,61,62);(2,4)(5,9)(6,15,16,41)(7,20)(8,25)(10,30)(11,32)(12,29)(13,34,35,48)(14,38,23,54)(17,42,18,44)(19,46)(21,51)(22,49)(24,55)(26,57)(27,58)(28,56)(31,50)(33,59,45,65)(36,60,43,61)(37,62,53,68)(39,63,40,64)(47,67)(52,66)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=i,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,h^b=hi,g^c=gi,h^c=hi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269702);