Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24808 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×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 | |
1382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,11,16,12,17,10,18)(7,19,24,37,23,44,22,13)(8,25,28,40,29,14,34,41)(9,30,32,42,33,43,31,15)(20,45,48,35,49,38,26,55)(21,50,53,64,52,36,51,46)(27,57,58,54,59,60,61,39)(47,66,67,65,68,62,56,63);(1,2,3,4)(5,7,11,24,12,23,10,22)(6,13,18,44,17,37,16,19)(8,26,34,49,29,48,28,20)(9,21,32,53,33,52,31,51)(14,38,41,55,25,45,40,35)(15,36,43,64,42,50,30,46)(27,56,61,68,59,67,58,47)(39,63,57,66,54,65,60,62);(2,4)(5,8)(6,14)(7,20)(9,27)(10,28)(11,34)(12,29)(13,35)(15,39)(16,40)(17,25)(18,41)(19,45)(21,47)(22,48)(23,49)(24,26)(30,60)(31,58)(32,61)(33,59)(36,62)(37,55)(38,44)(42,54)(43,57)(46,65)(50,66)(51,67)(52,68)(53,56)(63,64);(5,9,11,32,12,33,10,31)(6,15,18,43,17,42,16,30)(7,21,24,53,23,52,22,51)(8,27,34,61,29,59,28,58)(13,36,44,64,37,50,19,46)(14,39,41,57,25,54,40,60)(20,47,26,56,49,68,48,67)(35,62,38,63,55,66,45,65)Polycyclic presentation:
2^9|a^2=h,b^2=efgh,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,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, 274291);