Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #400055 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14795 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1333 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,19,33)(4,10,24,37)(5,25,27,11)(6,13,7,12)(9,20,32,18)(16,42,44,34)(17,41,46,57)(21,49,48,30)(22,35,52,55)(23,40,29,36)(26,38,28,39)(31,45,54,47)(43,58,63,61)(50,56,62,64)(51,60,53,59);(1,3)(2,9)(4,21)(5,17)(6,20)(7,18)(8,19)(10,34)(11,31)(12,33)(13,15)(14,32)(16,23)(22,50)(24,48)(25,54)(26,47)(27,46)(28,45)(29,44)(30,36)(35,58)(37,42)(38,41)(39,57)(40,49)(43,53)(51,63)(52,62)(55,61)(56,60)(59,64);(1,4)(2,10)(3,16)(5,22)(6,23)(7,29)(8,24)(9,30)(11,35)(12,36)(13,40)(14,37)(15,34)(17,43)(18,21)(19,44)(20,48)(25,55)(26,51)(27,52)(28,53)(31,56)(32,49)(33,42)(38,59)(39,60)(41,58)(45,62)(46,63)(47,50)(54,64)(57,61);(1,5,6,28,8,27,7,26)(2,11,12,39,14,25,13,38)(3,17,20,45,19,46,18,47)(4,22,23,53,24,52,29,51)(9,31,33,57,32,54,15,41)(10,35,36,60,37,55,40,59)(16,43,48,62,44,63,21,50)(30,56,42,61,49,64,34,58)Polycyclic presentation:
2^9|a^2=i,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 307252);