Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,8)(4,6)(5,7)(10,13)(14,16)(17,18,22,28,23,26,21,27)(19,24,30,39,31,38,35,41)(20,32,33,40,34,42,36,25)(29,43,45,47,46,48,44,37);(1,3,5,4)(2,6,7,8)(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,29)(21,35)(22,30)(23,31)(25,37)(26,38)(27,41)(28,39)(32,43)(33,45)(34,46)(36,44)(40,47)(42,48);(10,13)(14,16)(17,20,23,34)(18,25,26,40)(19,29,31,46)(21,33,22,36)(24,37,38,47)(27,32,28,42)(30,44,35,45)(39,48,41,43);(1,4,5,3)(2,6,7,8)(11,15)(14,16)(18,26)(24,38)(25,40)(27,28)(32,42)(37,47)(39,41)(43,48);(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,27,26,28)(19,30,31,35)(20,33,34,36)(24,39,38,41)(25,32,40,42)(29,44,46,45)(37,48,47,43)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ei,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6454594);