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 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1639 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18,23,28)(19,29,33,39)(20,34,36,41)(21,27,22,26)(24,32,40,30)(25,35,42,37)(31,45,46,38)(43,47,48,44);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(19,30)(20,35)(21,22)(24,29)(25,41)(26,27)(32,33)(34,42)(36,37)(39,40)(43,48)(44,47);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,19,21,30,23,33,22,32)(18,24,26,29,28,40,27,39)(20,31,37,44,36,46,35,47)(25,38,34,43,42,45,41,48);(9,11,12,16)(10,14,13,15)(17,20,21,37,23,36,22,35)(18,25,26,34,28,42,27,41)(19,31,30,44,33,46,32,47)(24,38,29,43,40,45,39,48);(1,4,5,3)(2,7,6,8)(20,36)(25,42)(31,46)(34,41)(35,37)(38,45)(43,48)(44,47)Polycyclic presentation:
2^9|a^2=i,c^2=fhi,d^2=g,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, 7504582);