Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #496344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #21794 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #3058 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
740 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)(4,19,23,51)(5,10)(6,12,27,38)(7,13)(8,16)(9,22)(11,26)(15,41,43,48)(17,40,47,61)(18,29)(20,52,44,59)(21,49,24,50)(25,35,57,37)(28,58)(30,62,63,45)(31,46)(32,64)(33,56)(34,55)(36,54)(39,42)(53,60);(1,3,5,16,7,18,26,46)(2,8,11,14,13,31,10,29)(4,20,21,43,23,44,24,15)(6,17,25,45,27,47,57,62)(9,32,33,60,34,39,36,28)(12,30,37,40,38,63,35,61)(19,48,50,52,51,41,49,59)(22,53,54,64,55,58,56,42);(1,4)(2,9)(3,15)(5,24)(6,22)(7,23)(8,28)(10,33)(11,36)(12,19)(13,34)(14,39)(16,44)(17,42)(18,43)(20,46)(21,26)(25,56)(27,55)(29,32)(30,59)(31,60)(35,50)(37,49)(38,51)(40,41)(45,58)(47,64)(48,61)(52,63)(53,62)(54,57);(2,10,13,11)(6,27)(8,29,31,14)(9,33,34,36)(12,37,38,35)(17,47)(19,49,51,50)(22,55)(25,57)(28,32,60,39)(30,40,63,61)(41,52,48,59)(42,64)(45,62)(53,58)(54,56)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=ce,e^c=ei,g^c=gh,f^d=fi,g^f=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403543);