Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480315 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#1799 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×M16 ◁ C2×#568 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1093 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)(3,13)(4,18)(5,22)(6,25,26,11)(7,12)(8,16)(9,21,30,48)(10,24)(14,36)(15,34)(17,43,41,35)(19,45)(20,49)(23,32,53,52)(27,55,56,64)(28,40)(29,42,54,63)(31,57,51,50)(33,38,58,62)(37,44)(39,60)(46,47)(59,61);(1,3,5,16,7,15,24,40)(2,8,10,13,12,28,22,34)(4,14,20,37,19,39,47,61)(6,17,23,42,26,41,53,63)(9,27,31,33,30,56,51,58)(11,29,32,35,25,54,52,43)(18,44,46,36,45,59,49,60)(21,38,50,55,48,62,57,64);(1,4)(2,9)(3,14)(5,20)(6,21)(7,19)(8,27)(10,31)(11,18)(12,30)(13,33)(15,39)(16,37)(17,38)(22,51)(23,50)(24,47)(25,45)(26,48)(28,56)(29,44)(32,46)(34,58)(35,36)(40,61)(41,62)(42,55)(43,60)(49,52)(53,57)(54,59)(63,64);(3,15)(4,19)(8,28)(9,30)(13,34)(16,40)(17,41)(18,45)(20,47)(21,48)(29,54)(31,51)(35,43)(42,63)(46,49)(50,57)Polycyclic presentation:
2^9|a^2=g,b^2=e,e^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=di,d^c=di,g^c=gh,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72303777, #72303801, #72303806, … |
| maximal normal subgroup of | #72303777, #72303801, #72303806, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400930);