Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #424740 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15324 |
| 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 | |
1044 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)(16,32)(17,39,47,43)(19,35)(20,34)(22,50)(23,52,55,49)(24,38)(26,37)(28,33,48,61)(29,40,31,41)(36,54,57,56)(42,46)(44,64,51,59)(45,60)(53,63,58,62);(1,3,8,20)(2,9,14,35)(4,22,26,16)(5,28,30,48)(6,19,7,18)(10,32,38,50)(11,39,27,43)(12,34,13,15)(17,31,47,29)(21,42,37,60)(23,53,55,58)(24,45,25,46)(33,41,61,40)(36,62,57,63)(44,54,51,56)(49,59,52,64);(1,4,7,25,8,26,6,24)(2,10,13,21,14,38,12,37)(3,16,19,46,20,22,18,45)(5,23,29,54,30,55,31,56)(9,32,34,60,35,50,15,42)(11,36,40,49,27,57,41,52)(17,44,28,58,47,51,48,53)(33,59,39,62,61,64,43,63);(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)(16,44)(18,48)(19,28)(20,47)(21,49)(22,51)(24,56)(25,54)(26,55)(32,59)(34,39)(35,61)(37,52)(38,57)(42,63)(45,53)(46,58)(50,64)(60,62)Polycyclic presentation:
2^9|a^2=ef,b^2=i,c^2=f,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,e^b=ei,f^b=fi,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, 331937);