Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419804 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15273 |
| 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 | |
1525 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)(3,15,18,35,19,34,30,9)(4,20,24,39,25,38,23,10)(5,26,28,11,29,41,31,42)(16,32,21,51,45,49,44,36)(17,46,27,57,48,33,47,40)(22,53,55,62,56,37,54,50)(43,60,52,59,63,64,58,61);(1,3)(2,9)(4,21)(5,27)(6,30)(7,18)(8,19)(10,36)(11,40)(12,34)(13,15)(14,35)(16,24)(17,28)(20,49)(22,43)(23,45)(25,44)(26,46)(29,47)(31,48)(32,38)(33,41)(37,61)(39,51)(42,57)(50,64)(52,54)(53,59)(55,58)(56,63)(60,62);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,25)(9,32)(11,37)(12,38)(13,20)(14,39)(15,36)(17,43)(18,44)(19,45)(21,30)(26,50)(27,58)(28,54)(29,56)(31,55)(33,59)(34,51)(35,49)(40,60)(41,62)(42,53)(46,61)(47,52)(48,63)(57,64);(1,5,6,28,8,29,7,31)(2,11,12,41,14,42,13,26)(3,17,18,27,19,48,30,47)(4,22,23,54,25,56,24,55)(9,33,15,40,35,46,34,57)(10,37,38,62,39,53,20,50)(16,43,44,58,45,63,21,52)(32,59,36,60,49,61,51,64)Polycyclic presentation:
2^9|a^2=f,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,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, 327000);