Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480314 |
| 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 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1095 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)(8,15)(9,21,30,48)(10,24)(14,36)(16,42,43,34)(17,35)(19,45)(20,49)(23,32,53,52)(27,55,56,64)(28,40,54,63)(29,41)(31,57,51,50)(33,39,58,62)(37,59)(38,44)(46,47)(60,61);(1,3,5,15,7,17,24,41)(2,8,10,13,12,29,22,35)(4,14,20,38,19,37,47,61)(6,16,23,40,26,43,53,63)(9,27,31,33,30,56,51,58)(11,28,32,34,25,54,52,42)(18,44,46,36,45,60,49,59)(21,39,50,55,48,62,57,64);(1,4)(2,9)(3,14)(5,20)(6,21)(7,19)(8,27)(10,31)(11,18)(12,30)(13,33)(15,38)(16,39)(17,37)(22,51)(23,50)(24,47)(25,45)(26,48)(28,44)(29,56)(32,46)(34,36)(35,58)(40,55)(41,61)(42,59)(43,62)(49,52)(53,57)(54,60)(63,64);(4,19)(9,30)(14,37)(18,45)(20,47)(21,48)(27,56)(31,51)(33,58)(36,59)(38,61)(39,62)(44,60)(46,49)(50,57)(55,64)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^c=di,g^c=gh,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72303781, #72303801, #72303804, … |
| maximal normal subgroup of | #72303781, #72303801, #72303804, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400948);