Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470028 |
| 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 | |
2913 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
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)(20,25)(21,28)(22,27)(23,26)(24,32)(30,46)(31,43)(33,45)(34,44)(35,41)(36,39)(37,42)(38,48)(40,47);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,19)(18,24)(20,33)(21,32)(22,36)(23,31)(25,40)(26,39)(27,29)(28,43)(30,35)(34,48)(37,47)(38,41)(42,46)(44,45);(3,9)(6,13)(10,11)(14,15)(17,20)(18,25)(19,30)(21,34)(22,37)(23,35)(24,38)(26,41)(27,42)(28,44)(29,45)(31,33)(32,47)(36,48)(39,40)(43,46);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(18,26)(19,31)(25,41)(27,28)(30,33)(32,36)(42,44)(47,48);(2,8)(6,13)(7,16)(14,15)(17,21,23,22)(18,27,26,28)(19,32,31,36)(20,34,35,37)(24,29,39,43)(25,42,41,44)(30,47,33,48)(38,45,40,46)Polycyclic presentation:
2^9|d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6388817);