Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437758 |
| 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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1466 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,12,15)(17,18)(19,29)(20,25)(21,26)(22,37)(23,28)(24,32)(27,34)(30,39)(31,43)(33,44)(35,41)(36,42)(38,47)(40,45)(46,48);(11,12)(14,15)(17,19,22,32,23,33,34,45)(18,24,27,29,28,40,37,44)(20,30,36,47,35,46,21,31)(25,38,26,39,41,43,42,48);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(17,20,23,35)(18,25,28,41)(19,30,33,46)(21,22,36,34)(24,38,40,43)(26,37,42,27)(29,39,44,48)(31,32,47,45);(9,12,13,11)(10,15,16,14)(17,21,23,36)(18,26,28,42)(19,31,33,47)(20,34,35,22)(24,39,40,48)(25,27,41,37)(29,43,44,38)(30,45,46,32);(1,3,4,7)(2,6,5,8)(20,35)(21,36)(25,41)(26,42)(30,46)(31,47)(38,43)(39,48)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=hi,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,d^c=di,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6375558);