Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#1867 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,11,7,12,24,34)(3,13,17,37,15,36,42,60)(4,9,20,31,18,29,44,55)(5,10,23,33,22,32,48,57)(8,25,28,41,27,50,54,16)(14,35,39,58,40,59,61,64)(19,30,45,56,21,46,47,62)(26,51,53,43,52,49,63,38);(1,3,6,17,7,15,24,42)(2,8,11,28,12,27,34,54)(4,14,20,39,18,40,44,61)(5,16,23,25,22,41,48,50)(9,26,31,53,29,52,55,63)(10,13,33,37,32,36,57,60)(19,38,45,51,21,43,47,49)(30,35,56,58,46,59,62,64);(1,4)(2,9)(3,14)(5,21)(6,20)(7,18)(8,26)(10,30)(11,31)(12,29)(13,35)(15,40)(16,43)(17,39)(19,22)(23,47)(24,44)(25,49)(27,52)(28,53)(32,46)(33,56)(34,55)(36,59)(37,58)(38,41)(42,61)(45,48)(50,51)(54,63)(57,62)(60,64);(3,15)(4,18)(8,27)(9,29)(13,36)(16,41)(17,42)(19,21)(20,44)(25,50)(28,54)(30,46)(31,55)(37,60)(45,47)(56,62)Polycyclic presentation:
2^9|a^2=g,b^2=g,g^2=i,b^a=be,c^a=cf,h^a=hi,d^b=di,f^b=fhi,d^c=di,e^c=eh,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72312701, #72312747, #72312796, … |
| maximal normal subgroup of | #72312701, #72312747, #72312796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401559);