Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470058 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2907 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18)(19,29)(20,34)(21,28)(22,27)(23,26)(24,33)(25,36)(30,39)(31,40)(32,41)(35,42)(37,44)(38,43)(45,46)(47,48);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,19)(18,24)(20,35)(21,32)(22,33)(23,31)(25,42)(26,41)(27,40)(28,29)(30,37)(34,45)(36,47)(38,46)(39,43)(44,48);(3,9)(6,13)(10,11)(14,15)(17,20)(18,25)(19,30)(21,36)(22,38)(23,37)(24,39)(26,43)(27,34)(28,44)(29,45)(31,35)(32,46)(33,47)(40,48)(41,42);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(18,26)(19,31)(25,43)(27,28)(30,35)(32,33)(34,44)(46,47);(2,8)(6,13)(7,16)(14,15)(17,21,23,22)(18,27,26,28)(19,32,31,33)(20,36,37,38)(24,40,41,29)(25,34,43,44)(30,46,35,47)(39,48,42,45)Polycyclic presentation:
2^9|d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6388795);