Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14517 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1026 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)(3,14,17,9,18,34,29,33)(4,19,22,10,23,37,21,38)(5,24,26,40,27,11,30,41)(15,31,45,50,47,35,46,42)(16,39,48,58,49,43,25,32)(20,51,53,36,55,61,54,56)(44,57,63,62,64,59,52,60);(1,3)(2,9)(4,15)(5,25)(6,29)(7,17)(8,18)(10,35)(11,39)(12,34)(13,33)(14,28)(16,26)(19,50)(20,52)(21,46)(22,45)(23,47)(24,43)(27,48)(30,49)(31,38)(32,40)(36,60)(37,42)(41,58)(44,53)(51,59)(54,64)(55,63)(56,62)(57,61);(1,4,7,22,8,23,6,21)(2,10,12,37,13,38,28,19)(3,15,17,45,18,47,29,46)(5,20,26,53,27,55,30,54)(9,31,34,50,33,35,14,42)(11,36,41,61,24,56,40,51)(16,44,48,63,49,64,25,52)(32,57,39,62,58,59,43,60);(1,5)(2,11)(3,16)(4,20)(6,30)(7,26)(8,27)(9,32)(10,36)(12,41)(13,24)(14,43)(15,44)(17,48)(18,49)(19,51)(21,54)(22,53)(23,55)(25,29)(28,40)(31,57)(33,58)(34,39)(35,59)(37,61)(38,56)(42,60)(45,63)(46,52)(47,64)(50,62)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295387);