Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| 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 | |
1012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2,4,3)(5,6,11,16,12,17,10,32)(7,18,23,36,22,45,21,13)(8,14,26,40,27,41,33,39)(9,28,30,42,31,58,29,15)(19,43,47,34,24,53,52,37)(20,48,51,65,50,35,49,44)(25,55,56,38,57,62,59,63)(46,61,66,64,54,60,67,68);(5,7)(6,13,17,36)(8,24,27,19)(9,20)(10,21)(11,23)(12,22)(14,37)(15,35,42,48)(16,18,32,45)(25,54,57,46)(26,52,33,47)(28,44,58,65)(29,49)(30,51)(31,50)(34,41)(38,61)(39,53)(40,43)(55,68)(56,67,59,66)(60,63)(62,64);(1,3,4,2)(5,8,10,26,12,27,11,33)(6,14,16,39,17,41,32,40)(7,19,21,47,22,24,23,52)(9,25,29,56,31,57,30,59)(13,34,18,43,36,37,45,53)(15,38,28,55,42,63,58,62)(20,46,49,66,50,54,51,67)(35,60,44,64,48,61,65,68);(5,9,11,30,12,31,10,29)(6,15,32,58,17,42,16,28)(7,20,23,51,22,50,21,49)(8,25,33,59,27,57,26,56)(13,35,45,65,36,48,18,44)(14,38,40,62,41,63,39,55)(19,46,52,67,24,54,47,66)(34,60,53,68,37,61,43,64)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fhi,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281960);