Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32950 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2847 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14698 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1749 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)(3,15,19,32,20,38,18,9)(4,21,24,35,25,34,23,10)(5,26,28,11,29,36,27,37)(16,41,44,58,45,30,43,39)(17,31,47,55,48,56,46,40)(22,50,52,49,53,33,51,57)(42,59,62,54,63,64,61,60);(1,3)(2,9,14,32)(4,16)(5,17)(6,18)(7,19)(8,20)(10,30,35,41)(11,31,37,56)(12,15,13,38)(21,39,34,58)(22,42)(23,43)(24,44)(25,45)(26,40,36,55)(27,46)(28,47)(29,48)(33,54,50,60)(49,59,57,64)(51,61)(52,62)(53,63);(1,4,6,23,8,25,7,24)(2,10,12,21,14,35,13,34)(3,16,18,43,20,45,19,44)(5,22,28,52,29,53,27,51)(9,30,15,39,32,41,38,58)(11,33,26,49,37,50,36,57)(17,42,47,62,48,63,46,61)(31,54,40,59,56,60,55,64);(1,5)(2,11)(3,17)(4,22)(6,28)(7,27)(8,29)(9,31)(10,33)(12,26)(13,36)(14,37)(15,40)(16,42)(18,47)(19,46)(20,48)(21,49)(23,52)(24,51)(25,53)(30,54)(32,56)(34,57)(35,50)(38,55)(39,59)(41,60)(43,62)(44,61)(45,63)(58,64)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^b=gi,e^d=ei,g^d=gi,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72281077, #72281086, #72281128 |
| maximal normal subgroup of | #72281077, #72281086, #72281128 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300289);