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 | #60450 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1769 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)(3,8,15,29)(4,17)(5,21,22,10)(6,24)(7,11)(9,19)(12,25)(13,33,35,56)(14,27)(16,40,39,28)(18,43,44,41)(20,42)(23,32,50,49)(26,51,53,36)(30,45,48,62)(31,46)(34,55)(37,59)(38,47)(52,57)(54,60)(58,61)(63,64);(1,3)(2,8,11,29)(4,13)(5,14)(6,16)(7,15)(9,26,31,53)(10,27,21,47)(12,28,24,40)(17,33,42,56)(18,34)(19,36)(20,35)(22,38)(23,37)(25,39)(30,52,48,63)(32,54,49,59)(41,55,43,61)(44,58)(45,57)(46,51)(50,60)(62,64);(1,4,6,19,7,20,25,46)(2,9,12,17,11,31,24,42)(3,13,16,36,15,35,39,51)(5,18,23,45,22,44,50,62)(8,26,28,33,29,53,40,56)(10,30,32,41,21,48,49,43)(14,34,37,57,38,58,60,64)(27,52,54,55,47,63,59,61);(1,5)(2,10)(3,14)(4,18)(6,23)(7,22)(8,27)(9,30)(11,21)(12,32)(13,34)(15,38)(16,37)(17,41)(19,45)(20,44)(24,49)(25,50)(26,52)(28,54)(29,47)(31,48)(33,55)(35,58)(36,57)(39,60)(40,59)(42,43)(46,62)(51,64)(53,63)(56,61)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=f,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gi,e^b=ei,g^b=gi,h^b=hi,e^d=ei,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72281077, #72281150, #72281304, … |
| maximal normal subgroup of | #72281077, #72281150, #72281304, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300229);