Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14187 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,6,13)(3,15,18,9,19,33,30,34)(4,20,23,38,25,37,24,10)(5,26,27,11,29,39,28,40)(16,31,21,49,44,57,43,35)(17,32,46,58,47,59,45,41)(22,51,53,55,54,36,52,48)(42,60,50,56,61,63,62,64);(1,3)(2,9)(4,21)(5,17)(6,30)(7,18)(8,19)(10,35)(11,32)(12,33)(13,15)(14,34)(16,24)(20,31)(22,50)(23,44)(25,43)(26,41)(27,46)(28,45)(29,47)(36,60)(37,57)(38,49)(39,58)(40,59)(42,52)(48,56)(51,63)(53,61)(54,62)(55,64);(1,4)(2,10)(3,16,19,44)(5,22)(6,24)(7,23)(8,25)(9,31,34,57)(11,36)(12,20)(13,37)(14,38)(15,35,33,49)(17,42,47,61)(18,21,30,43)(26,55)(27,53)(28,52)(29,54)(32,56,59,64)(39,48)(40,51)(41,60,58,63)(45,62,46,50);(1,5,7,27,8,29,6,28)(2,11,12,39,14,40,13,26)(3,17,18,46,19,47,30,45)(4,22,23,53,25,54,24,52)(9,32,33,58,34,59,15,41)(10,36,20,48,38,51,37,55)(16,42,21,50,44,61,43,62)(31,56,49,63,57,64,35,60)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=h,e^2=i,f^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,e^b=ei,g^b=gi,e^c=ei,f^c=fi,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 284148);