Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1311 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,25)(21,26)(22,23)(24,33)(27,29)(30,43)(31,37)(32,41)(34,38)(35,40)(36,42)(39,45)(44,48)(46,47);(1,3,5,8)(2,6,4,7)(9,11,13,10)(12,14,16,15)(18,23)(20,34)(24,37)(26,27)(28,41)(30,44)(35,36)(45,46);(1,2,5,4)(3,6,8,7)(10,11)(12,16)(17,19,29,31,22,32,21,33)(18,24,26,41,23,37,27,28)(20,30,35,46,34,44,36,45)(25,39,42,48,38,47,40,43);(1,4,5,2)(3,6,8,7)(9,12,13,16)(10,14,11,15)(17,20,22,34)(18,25,23,38)(19,30,32,44)(21,36,29,35)(24,39,37,47)(26,42,27,40)(28,43,41,48)(31,46,33,45);(10,11)(14,15)(17,21,22,29)(18,26,23,27)(19,31,32,33)(20,35,34,36)(24,28,37,41)(25,40,38,42)(30,45,44,46)(39,48,47,43)Polycyclic presentation:
2^9|a^2=fhi,b^2=fi,c^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6952496);