Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23990 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2404 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14016 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1361 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,27)(3,9,17,31,18,30,16,45)(4,10,23,35,21,34,22,38)(5,11,26,36,24,37,25,52)(14,28,41,32,40,54,19,46)(15,29,44,55,42,56,43,63)(20,48,51,59,49,33,50,58)(39,60,62,64,61,53,47,57);(1,3,7,17,8,18,6,16)(2,9,13,31,12,30,27,45)(4,19,23,14,21,41,22,40)(5,15,26,44,24,42,25,43)(10,32,35,54,34,46,38,28)(11,29,36,55,37,56,52,63)(20,47,51,39,49,62,50,61)(33,57,58,60,48,64,59,53);(1,4,6,23,8,21,7,22)(2,10,13,38,12,34,27,35)(3,14,16,41,18,40,17,19)(5,20,25,51,24,49,26,50)(9,28,31,46,30,54,45,32)(11,33,36,59,37,48,52,58)(15,39,43,62,42,61,44,47)(29,53,55,64,56,60,63,57);(1,5)(2,11)(3,15)(4,20)(6,25)(7,26)(8,24)(9,29)(10,33)(12,37)(13,36)(14,39)(16,43)(17,44)(18,42)(19,47)(21,49)(22,50)(23,51)(27,52)(28,53)(30,56)(31,55)(32,57)(34,48)(35,58)(38,59)(40,61)(41,62)(45,63)(46,64)(54,60)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=eg,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275434);