Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359650 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13867 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62859 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3707 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,26)(3,14,17,9,18,33,27,32)(4,19,21,36,22,10,29,37)(5,23,24,11,25,39,28,38)(15,40,45,34,44,58,43,30)(16,31,48,56,47,57,46,41)(20,50,52,35,53,54,51,49)(42,55,61,60,62,59,63,64);(1,3)(2,9)(4,15)(5,16)(6,27)(7,17)(8,18)(10,34)(11,31)(12,33)(13,32)(14,26)(19,30)(20,42)(21,45)(22,44)(23,41)(24,48)(25,47)(28,46)(29,43)(35,59)(36,40)(37,58)(38,57)(39,56)(49,55)(50,60)(51,63)(52,61)(53,62)(54,64);(1,4)(2,10)(3,15)(5,20)(6,21)(7,29)(8,22)(9,30)(11,35)(12,36)(13,19)(14,40)(16,42)(17,43)(18,44)(23,54)(24,51)(25,53)(26,37)(27,45)(28,52)(31,55)(32,34)(33,58)(38,49)(39,50)(41,60)(46,61)(47,62)(48,63)(56,64)(57,59);(1,5,8,25)(2,11,13,38)(3,16,18,47)(4,20,22,53)(6,28,7,24)(9,31,32,57)(10,35,19,49)(12,39,26,23)(14,41,33,56)(15,42,44,62)(17,48,27,46)(21,52,29,51)(30,55,34,59)(36,50,37,54)(40,60,58,64)(43,63,45,61)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72303081, #72303161, #72303171, … |
| maximal normal subgroup of | #72303081, #72303161, #72303171, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266849);