Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #470941 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16719 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
669 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)(3,13)(4,18,21,9)(5,10)(6,25)(7,12)(8,16)(11,27)(14,38,44,61)(15,45)(17,40)(19,32,52,50)(20,33,54,51)(22,55,53,34)(23,37)(24,36)(26,35)(28,56,31,43)(29,47)(30,41)(39,48)(42,62,63,58)(46,60,64,57)(49,59);(1,3)(2,8,12,30)(4,14,21,44)(5,15)(6,16)(7,17)(9,31)(10,29,36,59)(11,13,25,40)(18,28)(19,42,52,63)(20,43,54,56)(22,46,53,64)(23,47)(24,48)(26,49)(27,41)(32,60)(33,61)(34,62)(35,45,37,39)(38,51)(50,57)(55,58);(1,4,6,20,7,21,27,54)(2,9,25,51,12,18,11,33)(3,14,41,56,17,44,16,43)(5,22,23,52,24,53,26,19)(8,28,13,38,30,31,40,61)(10,34,37,32,36,55,35,50)(15,46,49,42,48,64,47,63)(29,58,45,60,59,62,39,57);(1,5,7,24)(2,10,12,36)(3,15,17,48)(4,19,21,52)(6,26,27,23)(8,29,30,59)(9,32,18,50)(11,37,25,35)(13,39,40,45)(14,42,44,63)(16,49,41,47)(20,53,54,22)(28,57,31,60)(33,55,51,34)(38,62,61,58)(43,64,56,46)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=ef,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378138);