Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| 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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1439 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(17,18,22,27)(19,28,32,40)(20,33,35,44)(21,25,29,26)(23,38,41,31)(24,36,43,37)(30,42,34,39)(45,46,48,47);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,29,38,22,32,21,31)(18,23,25,40,27,41,26,28)(20,34,36,47,35,30,37,46)(24,42,44,45,43,39,33,48);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,29,36,22,35,21,37)(18,24,25,44,27,43,26,33)(19,30,38,46,32,34,31,47)(23,39,40,48,41,42,28,45);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(20,35)(24,43)(30,34)(33,44)(36,37)(39,42)(45,48)(46,47);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(17,21,22,29)(18,25,27,26)(19,31,32,38)(20,36,35,37)(23,40,41,28)(24,33,43,44)(30,46,34,47)(39,45,42,48)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=fi,d^2=fgi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^c=di,e^c=ei,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6461459);