Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23872 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14171 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1623 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,28)(3,14,17,9,18,33,29,32)(4,19,22,10,23,36,21,37)(5,24,25,11,27,38,26,39)(15,30,43,49,45,34,44,40)(16,31,47,56,48,57,46,41)(20,51,52,50,53,35,54,59)(42,61,63,58,64,60,62,55);(1,3)(2,9)(4,15)(5,16)(6,29)(7,17)(8,18)(10,34)(11,31)(12,33)(13,32)(14,28)(19,49)(20,42)(21,44)(22,43)(23,45)(24,41)(25,47)(26,46)(27,48)(30,37)(35,58)(36,40)(38,56)(39,57)(50,61)(51,55)(52,63)(53,64)(54,62)(59,60);(1,4)(2,10)(3,15)(5,20)(6,21)(7,22)(8,23)(9,30)(11,35)(12,36)(13,37)(14,40)(16,42)(17,43)(18,45)(19,28)(24,50)(25,52)(26,54)(27,53)(29,44)(31,55)(32,34)(33,49)(38,59)(39,51)(41,60)(46,62)(47,63)(48,64)(56,61)(57,58);(1,5,7,25,8,27,6,26)(2,11,12,38,13,39,28,24)(3,16,17,47,18,48,29,46)(4,20,22,52,23,53,21,54)(9,31,33,56,32,57,14,41)(10,35,36,59,37,51,19,50)(15,42,43,63,45,64,44,62)(30,55,49,61,34,58,40,60)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,f^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,e^b=ei,f^b=fi,f^c=fi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72317339 |
| maximal normal subgroup of | #72317339 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283034);