Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #397514 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,3)(5,6,12,17,11,16,10,30)(7,13,21,48,20,37,31,36)(8,22,32,40,26,51,25,14)(9,27,33,41,29,57,28,15)(18,42,44,34,23,49,53,38)(19,45,47,35,46,65,58,61)(24,54,59,67,56,39,55,50)(43,62,64,63,52,60,68,66);(1,2,4,3)(5,7,10,21,11,20,12,31)(6,13,17,36,16,37,30,48)(8,23,25,53,26,18,32,44)(9,19,28,47,29,46,33,58)(14,38,22,49,40,34,51,42)(15,35,27,45,41,61,57,65)(24,52,55,68,56,43,59,64)(39,62,50,66,54,60,67,63);(1,3,4,2)(5,8)(6,14,16,40)(7,18,20,23)(9,24)(10,25)(11,26)(12,32)(13,34)(15,39,41,54)(17,22,30,51)(19,43,46,52)(21,44,31,53)(27,50,57,67)(28,55)(29,56)(33,59)(35,60)(36,42)(37,38)(45,63)(47,64,58,68)(48,49)(61,62)(65,66);(5,9,12,33,11,29,10,28)(6,15,30,57,16,41,17,27)(7,19,31,58,20,46,21,47)(8,24,32,59,26,56,25,55)(13,35,48,65,37,61,36,45)(14,39,51,67,40,54,22,50)(18,43,53,68,23,52,44,64)(34,60,49,66,38,62,42,63)Polycyclic presentation:
2^9|a^2=efi,b^2=f,c^2=e,d^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 320291);