Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6349370 |
| 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 | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2754 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)(3,9)(4,12)(5,7)(6,11)(8,13)(10,16)(14,15)(17,18)(19,28)(20,25)(21,27)(22,36)(23,26)(24,31)(29,42)(30,38)(32,41)(33,40)(34,39)(35,46)(37,45)(43,48)(44,47);(4,13)(8,12)(9,16)(11,14)(17,19,23,32)(18,24,26,38)(20,29,33,43)(21,31,22,30)(25,37,40,47)(27,41,36,28)(34,45,35,44)(39,48,46,42);(1,3)(2,6)(4,14)(5,10)(7,15)(8,16)(9,12)(11,13)(17,20)(18,25)(19,29)(21,34)(22,35)(23,33)(24,37)(26,40)(27,39)(28,42)(30,44)(31,45)(32,43)(36,46)(38,47)(41,48);(2,7)(6,15)(8,12)(9,16)(18,26)(20,33)(24,38)(27,36)(28,41)(29,43)(34,35)(44,45);(17,21,23,22)(18,27,26,36)(19,30,32,31)(20,34,33,35)(24,28,38,41)(25,39,40,46)(29,44,43,45)(37,42,47,48)Polycyclic presentation:
2^9|b^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256573);