Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419839 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2,3,4)(5,6,10,17,12,16,11,34)(7,18,21,43,23,13,22,37)(8,24,27,40,29,44,28,14)(9,30,31,60,33,15,32,42)(19,46,48,35,49,38,25,45)(20,41,50,66,52,36,51,62)(26,55,56,39,58,53,57,59)(47,64,67,61,68,63,54,65);(5,7)(6,13)(8,25)(9,20)(10,22)(11,21)(12,23)(14,38)(15,41)(16,18)(17,43)(19,28)(24,35)(26,54)(27,49)(29,48)(30,36)(31,51)(32,50)(33,52)(34,37)(39,64)(40,46)(42,62)(44,45)(47,57)(53,65)(55,61)(56,68)(58,67)(59,63)(60,66);(2,4)(5,8,12,29)(6,14,16,40)(7,19,23,49)(9,26,33,58)(10,28,11,27)(13,35,18,45)(15,39,30,59)(17,44,34,24)(20,47,52,68)(21,25,22,48)(31,57,32,56)(36,61,41,65)(37,46,43,38)(42,55,60,53)(50,54,51,67)(62,64,66,63);(5,9,12,33)(6,15)(7,20,23,52)(8,26,29,58)(10,31,11,32)(13,36)(14,39)(16,30)(17,42)(18,41)(19,47,49,68)(21,50,22,51)(24,53)(25,54,48,67)(27,56,28,57)(34,60)(35,61)(37,62)(38,63)(40,59)(43,66)(44,55)(45,65)(46,64)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327038);