Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6442587 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(5,6,12,18)(7,19,23,42)(8,24,28,37)(9,15,29,39)(10,17,11,16)(13,32,35,22)(14,26,38,27)(20,45,48,33)(21,44,49,62)(25,51,52,60)(30,41,31,40)(34,56,58,50)(36,53,59,54)(43,55,61,47)(46,64,65,57)(63,67,68,66);(1,2)(3,4)(5,7)(6,13)(8,20)(9,21)(10,32)(11,22)(12,23)(14,33)(15,34)(16,42)(17,19)(18,35)(24,43)(25,46)(26,55)(27,47)(28,48)(29,49)(30,56)(31,50)(36,57)(37,61)(38,45)(39,58)(40,62)(41,44)(51,63)(52,65)(53,67)(54,66)(59,64)(60,68);(2,4)(5,8,10,26,12,28,11,27)(6,14,16,37,18,38,17,24)(7,20,32,55,23,48,22,47)(9,25,30,53,29,52,31,54)(13,33,42,61,35,45,19,43)(15,36,40,60,39,59,41,51)(21,46,56,67,49,65,50,66)(34,57,62,68,58,64,44,63);(2,4)(5,9)(6,15)(7,21)(8,25)(10,30)(11,31)(12,29)(13,34)(14,36)(16,40)(17,41)(18,39)(19,44)(20,46)(22,50)(23,49)(24,51)(26,53)(27,54)(28,52)(32,56)(33,57)(35,58)(37,60)(38,59)(42,62)(43,63)(45,64)(47,66)(48,65)(55,67)(61,68);(9,29)(15,39)(21,49)(25,52)(30,31)(34,58)(36,59)(40,41)(44,62)(46,65)(50,56)(51,60)(53,54)(57,64)(63,68)(66,67)Polycyclic presentation:
2^9|a^2=i,c^2=gi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349785);