Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460711 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16370 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
891 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)(3,14,18,32)(4,19)(5,10)(6,11)(7,13)(8,17,33,52)(9,23)(12,29)(15,34,20,30)(16,47,50,43)(21,37)(22,36)(24,41)(25,58)(26,38)(27,40)(28,39)(31,49,62,51)(35,56)(42,45,53,46)(44,63,55,60)(48,64,59,61)(54,57);(1,3,7,18)(2,8)(4,20)(5,16,27,50)(6,17,29,52)(9,34,36,30)(10,31)(11,14)(12,32)(13,33)(15,24)(19,53,41,42)(21,55)(22,46)(23,45)(25,59)(26,49,28,51)(35,63,58,60)(37,64,54,61)(38,47)(39,43)(40,62)(44,57)(48,56);(1,4,6,22,7,24,29,23)(2,9,11,19,13,36,12,41)(3,15,17,45,18,20,52,46)(5,25,26,57,27,56,28,21)(8,30,14,42,33,34,32,53)(10,37,38,58,40,54,39,35)(16,48,49,55,50,59,51,44)(31,61,47,63,62,64,43,60);(1,5,7,27)(2,10,13,40)(3,16,18,50)(4,21,24,57)(6,28,29,26)(8,31,33,62)(9,35,36,58)(11,39,12,38)(14,43,32,47)(15,44,20,55)(17,51,52,49)(19,54,41,37)(22,56,23,25)(30,60,34,63)(42,64,53,61)(45,59,46,48)Polycyclic presentation:
2^9|a^2=g,b^2=ef,c^2=e,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367908);