Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359901 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2342 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13883 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
969 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,14)(3,15,18,33,20,9,19,34)(4,21,25,37,26,40,24,10)(5,27,28,38,29,11,30,39)(16,35,22,41,44,57,48,31)(17,32,46,58,47,42,45,59)(23,51,52,55,54,49,53,36)(43,62,63,61,64,60,50,56);(1,3,6,18,8,20,7,19)(2,9,14,34,12,15,13,33)(4,22,24,44,26,48,25,16)(5,17,28,45,29,47,30,46)(10,35,40,41,37,57,21,31)(11,32,39,59,27,42,38,58)(23,50,52,64,54,63,53,43)(36,60,51,61,55,62,49,56);(1,4,6,24,8,26,7,25)(2,10,13,21,12,37,14,40)(3,16,18,22,20,44,19,48)(5,23,28,52,29,54,30,53)(9,31,33,57,15,41,34,35)(11,36,38,49,27,55,39,51)(17,43,45,50,47,64,46,63)(32,56,58,62,42,61,59,60);(1,5)(2,11)(3,17)(4,23)(6,28)(7,30)(8,29)(9,32)(10,36)(12,27)(13,38)(14,39)(15,42)(16,43)(18,45)(19,46)(20,47)(21,49)(22,50)(24,52)(25,53)(26,54)(31,56)(33,58)(34,59)(35,60)(37,55)(40,51)(41,61)(44,64)(48,63)(57,62)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268101);