Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470237 |
| 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 | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3021 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)(19,23)(20,24)(21,33)(22,25)(26,34)(27,38)(28,37)(29,43)(30,35)(31,45)(32,39)(36,42)(40,46)(41,47)(44,48);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,19)(18,23)(20,30)(21,29)(22,28)(24,38)(25,37)(26,36)(27,32)(31,44)(33,43)(34,42)(35,39)(40,48)(41,46)(45,47);(3,9)(6,13)(10,11)(14,15)(17,20)(18,24)(19,27)(21,31)(22,32)(23,35)(25,39)(26,40)(28,30)(29,41)(33,45)(34,46)(36,47)(37,38)(42,44)(43,48);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(18,25)(19,28)(24,39)(26,33)(27,30)(29,42)(40,45)(41,44);(2,8)(6,13)(7,16)(14,15)(17,21,22,34)(18,26,25,33)(19,29,28,42)(20,31,32,46)(23,36,37,43)(24,40,39,45)(27,41,30,44)(35,47,38,48)Polycyclic presentation:
2^9|d^2=hi,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dh,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, 6388665);