Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #392138 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14679 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×2−1+6 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
15641 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)(4,18,21,9)(5,10)(6,24)(7,12)(8,16)(11,27)(14,39,42,37)(15,43)(17,35)(19,48,51,31)(20,32,54,49)(22,36)(23,34)(25,30)(26,33)(28,41,47,52)(29,44)(38,45)(40,59,62,60)(46,56)(50,57,53,58)(55,61,64,63);(1,3)(2,8)(4,14)(5,15)(6,25)(7,17)(9,28)(10,29)(11,35)(12,30)(13,24)(16,27)(18,47)(19,40)(20,52)(21,42)(22,46)(23,45)(26,44)(31,55)(32,39)(33,38)(34,56)(36,43)(37,49)(41,54)(48,64)(50,63)(51,62)(53,61)(57,59)(58,60);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,28)(10,31)(11,32)(12,18)(13,37)(15,40)(16,41)(17,42)(22,50)(23,51)(24,49)(25,52)(26,53)(27,54)(29,55)(30,47)(33,57)(34,48)(35,39)(36,58)(38,59)(43,60)(44,61)(45,62)(46,63)(56,64);(1,5,7,23)(2,10,12,34)(3,15,17,45)(4,19,21,51)(6,26,27,22)(8,29,30,56)(9,31,18,48)(11,36,24,33)(13,38,35,43)(14,40,42,62)(16,46,25,44)(20,53,54,50)(28,55,47,64)(32,58,49,57)(37,59,39,60)(41,63,52,61)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,e^b=ei,g^b=gi,h^c=hi,e^d=ei89 conjugacy classes — view the full character table.
| maximal subgroup of | #72262954, #72263037, #72263043, … |
| maximal normal subgroup of | #72262954, #72263037, #72263043, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299335);