Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13030 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1879 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #10788 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
951 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,7,13,8,14,31,15)(3,16,20,45,22,38,21,43)(4,23,25,46,27,57,26,44)(5,28,32,11,6,33,36,12)(9,29,39,19,40,34,47,18)(10,30,41,58,42,35,48,24)(17,50,52,62,53,37,56,61)(49,55,64,59,63,54,60,51);(1,3)(2,9,14,40)(4,17)(5,29)(6,34)(7,20)(8,22)(10,37,42,50)(11,43,12,45)(13,39,15,47)(16,28,38,33)(18,36)(19,32)(21,31)(23,49,57,63)(24,54)(25,52)(26,56)(27,53)(30,51)(35,59)(41,61,48,62)(44,60,46,64)(55,58);(1,4)(2,10)(3,17)(5,30)(6,35)(7,25)(8,27)(9,37)(11,44)(12,46)(13,41)(14,42)(15,48)(16,49)(18,54)(19,55)(20,52)(21,56)(22,53)(23,33)(24,36)(26,31)(28,57)(29,51)(32,58)(34,59)(38,63)(39,61)(40,50)(43,60)(45,64)(47,62)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,d^b=dhi,e^b=ehi,h^b=hi,d^c=dhi,e^c=eh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72279896, #72280192, #72281128, … |
| maximal normal subgroup of | #72279896, #72280192, #72281128, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53961);