Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432876 |
| 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 | |
2933 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)(4,8)(5,7)(9,12)(10,13)(11,14)(15,16)(17,18)(19,29)(20,25)(21,28)(22,26)(23,27)(24,31)(30,46)(32,43)(33,45)(34,44)(35,42)(36,39)(37,41)(38,48)(40,47);(1,3)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16)(17,19)(18,24)(20,33)(21,31)(22,36)(23,32)(25,40)(26,29)(27,39)(28,43)(30,35)(34,48)(37,47)(38,42)(41,46)(44,45);(3,9)(6,12)(10,15)(13,16)(17,20)(18,25)(19,30)(21,34)(22,37)(23,35)(24,38)(26,41)(27,42)(28,44)(29,45)(31,47)(32,33)(36,48)(39,40)(43,46);(2,7)(6,12)(8,14)(13,16)(17,21,23,22)(18,26,27,28)(19,31,32,36)(20,34,35,37)(24,29,39,43)(25,41,42,44)(30,47,33,48)(38,45,40,46);(1,4,5,11)(2,8,7,14)(3,10,9,15)(6,13,12,16)(18,27)(24,39)(25,42)(26,28)(29,43)(38,40)(41,44)(45,46)Polycyclic presentation:
2^9|d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fi,c^b=ch,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, 6343813);