Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #425084 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15328 |
| 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 | |
988 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14,7,13,6,12)(3,15,30,9,19,35,18,34)(4,20,25,38,24,10,23,42)(5,26,28,11,29,40,31,41)(16,45,48,36,21,43,47,32)(17,49,27,44,50,33,51,39)(22,54,56,37,57,63,58,52)(46,61,59,62,53,60,55,64);(1,3,7,19)(2,9,13,34)(4,21,24,16)(5,27,29,51)(6,30,8,18)(10,36,20,32)(11,39,41,44)(12,35,14,15)(17,31,50,28)(22,55,57,59)(23,47,25,48)(26,49,40,33)(37,62,52,64)(38,43,42,45)(46,56,53,58)(54,60,63,61);(1,4,8,25,7,24,6,23)(2,10,14,42,13,20,12,38)(3,16,30,48,19,21,18,47)(5,22,28,56,29,57,31,58)(9,32,35,45,34,36,15,43)(11,37,40,63,41,52,26,54)(17,46,27,59,50,53,51,55)(33,60,39,64,49,61,44,62);(1,5)(2,11)(3,17)(4,22)(6,31)(7,29)(8,28)(9,33)(10,37)(12,26)(13,41)(14,40)(15,44)(16,46)(18,51)(19,50)(20,52)(21,53)(23,58)(24,57)(25,56)(27,30)(32,60)(34,49)(35,39)(36,61)(38,54)(42,63)(43,62)(45,64)(47,55)(48,59)Polycyclic presentation:
2^9|a^2=ef,b^2=i,c^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 332281);