Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24808 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2416 |
| 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 | |
1622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,4)(5,6,11,16,12,18,10,17)(7,19,23,38,24,37,22,13)(8,25,28,14,30,41,29,42)(9,31,33,44,34,43,32,15)(20,47,26,45,50,35,49,39)(21,51,53,62,54,36,52,46)(27,57,58,55,60,40,59,64)(48,63,56,66,68,65,67,61);(5,7,10,22,12,24,11,23)(6,13,16,19,18,38,17,37)(8,26,28,50,30,49,29,20)(9,21,32,52,34,54,33,53)(14,39,25,35,42,45,41,47)(15,36,31,46,44,51,43,62)(27,56,58,68,60,67,59,48)(40,63,55,61,57,65,64,66);(1,2,3,4)(5,8)(6,14)(7,20)(9,27)(10,28)(11,29)(12,30)(13,35)(15,40)(16,25)(17,41)(18,42)(19,45)(21,48)(22,26)(23,49)(24,50)(31,55)(32,58)(33,59)(34,60)(36,61)(37,39)(38,47)(43,64)(44,57)(46,65)(51,66)(52,56)(53,67)(54,68)(62,63);(5,9,11,33,12,34,10,32)(6,15,17,43,18,44,16,31)(7,21,23,53,24,54,22,52)(8,27,29,59,30,60,28,58)(13,36,37,62,38,51,19,46)(14,40,41,64,42,57,25,55)(20,48,49,67,50,68,26,56)(35,61,39,63,47,66,45,65)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fh,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281648);