Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24987 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2420 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13970 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1581 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)(3,15,19,33,20,39,18,9)(4,21,24,10,26,36,25,35)(5,27,29,38,30,37,28,11)(16,31,43,57,22,50,48,40)(17,44,46,59,47,32,45,41)(23,52,53,49,55,34,54,58)(42,60,61,56,51,64,63,62);(1,3)(2,9,14,33)(4,22,26,16)(5,17)(6,18)(7,19)(8,20)(10,31)(11,32,38,44)(12,15,13,39)(21,40)(23,51,55,42)(24,48,25,43)(27,41,37,59)(28,45)(29,46)(30,47)(34,56)(35,50)(36,57)(49,60)(52,62)(53,63,54,61)(58,64);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,26)(9,31)(11,34)(12,21)(13,36)(14,35)(15,40)(17,42)(18,43)(19,48)(20,22)(27,49)(28,53)(29,54)(30,55)(32,56)(33,50)(37,58)(38,52)(39,57)(41,60)(44,62)(45,61)(46,63)(47,51)(59,64);(1,5,6,28,8,30,7,29)(2,11,12,27,14,38,13,37)(3,17,18,45,20,47,19,46)(4,23,24,53,26,55,25,54)(9,32,15,41,33,44,39,59)(10,34,21,49,35,52,36,58)(16,42,43,61,22,51,48,63)(31,56,40,60,50,62,57,64)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272244);