Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427377 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15369 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1204 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)(3,8)(4,9)(5,21,23,10)(6,25)(7,12)(11,28)(13,41)(14,30,22,37)(15,31)(16,39)(17,33)(18,47,49,34)(19,40)(20,36)(24,38,54,51)(26,32)(27,35)(29,43)(42,62,52,56)(44,63,53,57)(45,58)(46,59,48,60)(50,61,55,64);(1,3)(2,8)(4,17)(5,22)(6,26)(7,15)(9,33)(10,37)(11,39)(12,31)(13,19)(14,23)(16,28)(18,48)(20,45)(21,30)(24,44)(25,32)(27,43)(29,35)(34,60)(36,58)(38,57)(40,41)(42,55)(46,49)(47,59)(50,52)(51,63)(53,54)(56,64)(61,62);(1,4,7,20)(2,9,12,36)(3,13,15,43)(5,18,23,49)(6,27,28,19)(8,29,31,41)(10,34,21,47)(11,40,25,35)(14,42,22,52)(16,45,26,17)(24,55,54,50)(30,56,37,62)(32,58,39,33)(38,64,51,61)(44,48,53,46)(57,60,63,59);(1,5)(2,10)(3,14,15,22)(4,18)(6,24)(7,23)(8,30,31,37)(9,34)(11,38)(12,21)(13,42,43,52)(16,44,26,53)(17,46,45,48)(19,50)(20,49)(25,51)(27,55)(28,54)(29,56,41,62)(32,57,39,63)(33,59,58,60)(35,61)(36,47)(40,64)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334574);