Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518920 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1697 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,26)(19,27,31,42)(20,32,34,46)(21,35,29,25)(23,36,38,28)(24,39,40,33)(30,44,45,37)(41,47,48,43);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(19,28)(20,33)(21,29)(23,27)(24,32)(25,35)(30,45)(31,36)(34,39)(37,44)(38,42)(40,46);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,19,29,36,22,31,21,28)(18,23,35,42,26,38,25,27)(20,30,39,47,34,45,33,43)(24,37,46,48,40,44,32,41);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20,21,33,22,34,29,39)(18,24,25,32,26,40,35,46)(19,30,28,43,31,45,36,47)(23,37,27,41,38,44,42,48);(1,4,5,3)(2,7,6,8)(20,34)(24,40)(30,45)(32,46)(33,39)(37,44)(41,48)(43,47)Polycyclic presentation:
2^9|a^2=fi,c^2=fhi,d^2=fg,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7505859);