Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13417 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2041 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #11008 |
| prime radical | #60643 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1121 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)(3,14)(4,20,22,37)(5,24)(6,30)(7,12)(8,16)(9,21,38,58)(10,28)(11,42)(13,27)(15,33,52,36)(17,34)(18,43)(19,49)(23,41,32,44)(25,46)(26,39,59,47)(29,45)(31,40)(35,56)(48,51,64,50)(53,62,57,61)(54,60,55,63);(1,3)(2,8)(4,15)(5,25)(6,31)(7,19)(9,36)(10,40)(11,43)(12,34)(13,46)(14,24)(16,28)(17,42)(18,27)(20,48)(21,54)(22,52)(23,57)(26,50)(29,56)(30,35)(32,53)(33,38)(37,64)(39,61)(41,63)(44,60)(45,49)(47,62)(51,59)(55,58);(1,4)(2,9)(3,15)(5,26)(6,32)(7,22)(8,33)(10,41)(11,44)(12,38)(13,47)(14,48)(16,54)(17,55)(18,57)(19,52)(20,45)(21,28)(23,27)(24,37)(25,50)(29,59)(30,39)(31,53)(34,36)(35,62)(40,60)(42,58)(43,63)(46,61)(49,64)(51,56)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,h^b=hi,d^c=dh,e^c=eh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279962, #72280442, #72307190, … |
| maximal normal subgroup of | #72279962, #72280442, #72307190, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55093);