Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219542 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| 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 | |
1252 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)(7,19)(8,25,30,41)(9,15,34,46)(10,17)(11,16)(12,18)(13,22)(14,29,42,28)(20,35,50,53)(21,47,52,43)(23,38)(24,37)(26,39,49,62)(27,56)(31,60,51,36)(32,45,33,44)(40,57)(48,64)(54,59)(55,66)(58,65)(61,68)(63,67);(5,7,12,24)(6,13,18,38)(8,26,30,49)(9,31,34,51)(10,23,11,22)(14,39,42,62)(15,43,46,47)(16,37,17,19)(20,29,50,28)(21,33,52,32)(25,53,41,35)(27,48,59,68)(36,45,60,44)(40,64,65,61)(54,66,56,63)(55,58,67,57);(2,4)(5,8,10,29,12,30,11,28)(6,14,16,41,18,42,17,25)(7,20,22,49,24,50,23,26)(9,27,33,57,34,59,32,58)(13,35,19,39,38,53,37,62)(15,40,45,56,46,65,44,54)(21,48,51,67,52,68,31,55)(36,61,47,63,60,64,43,66);(2,4)(5,9)(6,15)(7,21)(8,27)(10,33)(11,32)(12,34)(13,36)(14,40)(16,45)(17,44)(18,46)(19,47)(20,48)(22,51)(23,31)(24,52)(25,54)(26,55)(28,58)(29,57)(30,59)(35,61)(37,43)(38,60)(39,63)(41,56)(42,65)(49,67)(50,68)(53,64)(62,66)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=eh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295586);