Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470402 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,7,8,6)(11,15)(14,16)(17,18,22,27)(19,28,33,44)(20,25,34,24)(21,36,29,26)(23,37,39,32)(30,46,31,43)(35,40,45,42)(38,47,41,48);(2,5)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,29,37,22,33,21,32)(18,23,36,44,27,39,26,28)(20,30,45,48,34,31,35,47)(24,38,42,46,25,41,40,43);(1,3,4,8)(2,6,5,7)(10,13)(14,16)(17,20,22,34)(18,24,27,25)(19,30,33,31)(21,35,29,45)(23,38,39,41)(26,40,36,42)(28,43,44,46)(32,47,37,48);(10,13)(14,16)(17,20)(18,25)(19,31)(21,35)(22,34)(23,38)(24,27)(26,42)(28,43)(29,45)(30,33)(32,48)(36,40)(37,47)(39,41)(44,46);(9,11,12,15)(10,14,13,16)(20,34)(24,25)(30,31)(35,45)(38,41)(40,42)(43,46)(47,48)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,d^c=di,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, 6402392);