Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419211 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15268 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3020 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1808 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)(3,15)(4,21)(5,27,30,11)(6,13)(7,12)(8,14)(9,18)(10,25)(16,44)(17,39,48,43)(19,35)(20,34)(22,32)(23,52,55,50)(24,38)(26,37)(28,33,49,60)(29,40,31,41)(36,54,57,56)(42,47)(45,58,51,63)(46,59)(53,61,64,62);(1,3,6,18,8,20,7,19)(2,9,12,15,14,35,13,34)(4,22,24,47,26,16,25,46)(5,28,29,48,30,49,31,17)(10,32,37,59,38,44,21,42)(11,39,40,60,27,43,41,33)(23,53,54,45,55,64,56,51)(36,61,50,63,57,62,52,58);(1,4,8,26)(2,10,14,38)(3,16,20,22)(5,23,30,55)(6,25,7,24)(9,32,35,44)(11,36,27,57)(12,21,13,37)(15,42,34,59)(17,45,48,51)(18,47,19,46)(28,53,49,64)(29,56,31,54)(33,58,60,63)(39,62,43,61)(40,52,41,50);(1,5)(2,11)(3,17)(4,23)(6,31)(7,29)(8,30)(9,33)(10,36)(12,41)(13,40)(14,27)(15,43)(16,45)(18,49)(19,28)(20,48)(21,50)(22,51)(24,56)(25,54)(26,55)(32,58)(34,39)(35,60)(37,52)(38,57)(42,61)(44,63)(46,53)(47,64)(59,62)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,e^d=ei,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326408);