Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #27987 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2504 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14050 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
703 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)(3,14)(4,20,25,38)(5,26)(6,12)(7,13)(8,17)(9,24,34,23)(10,27)(11,29)(15,30,21,49)(16,43)(18,32)(19,37)(22,51,54,33)(28,35)(31,46)(36,55)(39,42,57,47)(40,44)(41,60,50,64)(45,58)(48,53,59,52)(56,63,62,61);(1,3)(2,8,13,32)(4,21)(5,16)(6,17)(7,19)(9,30,34,49)(10,31,36,43)(11,14,12,37)(15,25)(18,29)(20,39,38,57)(22,50)(23,47)(24,42)(26,40,35,58)(27,44)(28,46)(33,56,51,62)(41,54)(45,55)(48,60,59,64)(52,63)(53,61);(1,4,7,25)(2,9,13,34)(3,15)(5,22,28,54)(6,23,29,24)(8,30)(10,33,36,51)(11,20,12,38)(14,39)(16,41)(17,42)(18,47)(19,21)(26,48,35,59)(27,52,55,53)(31,56)(32,49)(37,57)(40,60)(43,62)(44,61)(45,63)(46,50)(58,64);(1,5,6,27,7,28,29,55)(2,10,11,26,13,36,12,35)(3,16,17,44,19,46,18,45)(4,22,23,52,25,54,24,53)(8,31,14,40,32,43,37,58)(9,33,20,48,34,51,38,59)(15,41,42,61,21,50,47,63)(30,56,39,60,49,62,57,64)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=efg,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 277637);