Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #424343 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15320 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,14,7,13)(3,15,18,35,20,34,19,9)(4,21,24,38,26,42,25,10)(5,27,29,11,31,40,30,41)(16,43,45,59,46,32,22,36)(17,47,28,56,49,33,48,39)(23,37,52,50,54,62,53,55)(44,63,57,58,64,60,51,61);(1,3)(2,9)(4,22)(5,28)(6,19)(7,18)(8,20)(10,36)(11,39)(12,34)(13,15)(14,35)(16,25)(17,29)(21,32)(23,44)(24,46)(26,45)(27,47)(30,49)(31,48)(33,40)(37,61)(38,59)(41,56)(42,43)(50,60)(51,52)(53,57)(54,64)(55,63)(58,62);(1,4)(2,10,14,38)(3,16)(5,23)(6,24)(7,25)(8,26)(9,32,35,43)(11,37,41,62)(12,21,13,42)(15,36,34,59)(17,44)(18,45)(19,22)(20,46)(27,55,40,50)(28,57)(29,52)(30,53)(31,54)(33,58,47,61)(39,60,56,63)(48,51)(49,64);(1,5,6,29,8,31,7,30)(2,11,12,40,14,41,13,27)(3,17,18,28,20,49,19,48)(4,23,24,52,26,54,25,53)(9,33,15,39,35,47,34,56)(10,37,21,50,38,62,42,55)(16,44,45,57,46,64,22,51)(32,58,36,60,43,61,59,63)Polycyclic presentation:
2^9|a^2=e,c^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331539);