Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1873 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(3,7)(6,8)(9,10,16,22)(11,23,26,50)(12,28,32,52)(13,19,36,49)(14,21,15,20)(17,37,42,27)(18,39,45,31)(24,40,29,43)(25,54,59,46)(30,61,34,66)(33,65,60,41)(35,51,38,48)(44,63,47,64)(53,56,69,57)(55,68,58,70)(62,72,67,71);(2,5)(3,7)(9,11)(10,17)(12,29)(13,33)(14,37)(15,27)(16,26)(18,43)(19,46)(20,50)(21,23)(22,42)(24,32)(25,35)(28,53)(30,62)(31,56)(34,67)(36,60)(38,59)(39,57)(40,45)(41,48)(44,71)(47,72)(49,54)(51,65)(52,69)(55,64)(58,63)(61,68)(66,70);(1,2)(3,6)(4,5)(7,8)(9,12,16,32)(10,18,22,45)(11,24,26,29)(13,34,36,30)(14,31,15,39)(17,40,42,43)(19,47,49,44)(20,28,21,52)(23,53,50,69)(25,58,59,55)(27,56,37,57)(33,62,60,67)(35,64,38,63)(41,70,65,68)(46,71,54,72)(48,61,51,66);(2,5)(6,8)(9,13,16,36)(10,19,22,49)(11,25,26,59)(12,30,32,34)(14,38,15,35)(17,41,42,65)(18,44,45,47)(20,51,21,48)(23,54,50,46)(24,55,29,58)(27,60,37,33)(28,61,52,66)(31,64,39,63)(40,68,43,70)(53,71,69,72)(56,62,57,67);(1,3,4,7)(2,6,5,8)(11,26)(17,42)(23,50)(24,29)(25,59)(27,37)(33,60)(40,43)(41,65)(46,54)(53,69)(55,58)(56,57)(62,67)(68,70)(71,72)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7050422);