Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454977 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
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 | |
1405 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)(9,30,31,59,33,15,32,42)(19,47,49,35,50,38,25,45)(20,36,51,62,53,46,52,66)(26,54,56,41,58,65,57,39)(48,64,67,60,68,63,55,61);(5,7,12,23)(6,13,16,18)(8,25,29,49)(9,20,33,53)(10,22,11,21)(14,38,40,47)(15,36,30,46)(17,43,34,37)(19,27,50,28)(24,35,44,45)(26,55,58,67)(31,52,32,51)(39,63,41,64)(42,66,59,62)(48,56,68,57)(54,60,65,61);(2,4)(5,8)(6,14)(7,19)(9,26)(10,28)(11,27)(12,29)(13,35)(15,41)(16,40)(17,44)(18,45)(20,48)(21,25)(22,49)(23,50)(24,34)(30,39)(31,57)(32,56)(33,58)(36,61)(37,47)(38,43)(42,54)(46,60)(51,55)(52,67)(53,68)(59,65)(62,63)(64,66);(5,9,12,33)(6,15)(7,20,23,53)(8,26,29,58)(10,31,11,32)(13,36)(14,39)(16,30)(17,42)(18,46)(19,48,50,68)(21,51,22,52)(24,54)(25,55,49,67)(27,56,28,57)(34,59)(35,60)(37,62)(38,63)(40,41)(43,66)(44,65)(45,61)(47,64)Polycyclic presentation:
2^9|a^2=f,b^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,f^b=fi,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362176);