Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393998 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14716 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2569 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,13)(4,9)(5,10)(6,23)(7,12)(8,16)(11,24)(14,37)(15,35)(17,36)(18,44)(19,47)(20,31)(21,49)(22,33)(25,51)(26,41)(27,43)(28,39)(29,46)(30,48)(32,50)(34,40)(38,56)(42,57)(45,55)(52,59)(53,61)(54,64)(58,60)(62,63);(1,3,6,16,7,17,24,43)(2,8,11,13,12,27,23,36)(4,14,19,39,20,40,48,51)(5,15,21,41,22,42,50,61)(9,28,30,37,31,25,47,34)(10,26,32,35,33,53,49,57)(18,38,45,59,46,60,63,64)(29,54,55,58,44,52,62,56);(1,4)(2,9)(3,14)(5,18)(6,19)(7,20)(8,25)(10,29)(11,30)(12,31)(13,34)(15,38)(16,39)(17,40)(21,45)(22,46)(23,47)(24,48)(26,52)(27,28)(32,55)(33,44)(35,56)(36,37)(41,59)(42,60)(43,51)(49,62)(50,63)(53,54)(57,58)(61,64);(1,5)(2,10)(3,15)(4,18)(6,21)(7,22)(8,26)(9,29)(11,32)(12,33)(13,35)(14,38)(16,41)(17,42)(19,45)(20,46)(23,49)(24,50)(25,52)(27,53)(28,54)(30,55)(31,44)(34,56)(36,57)(37,58)(39,59)(40,60)(43,61)(47,62)(48,63)(51,64)Polycyclic presentation:
2^9|b^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301401);