Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324500 |
| 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 | #45316 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1484 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)(17,18)(19,23)(20,31)(21,25)(22,27)(24,28)(26,30)(29,38)(32,40)(33,44)(34,37)(35,41)(36,42)(39,43)(45,48)(46,47);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,27,26)(20,32,34,28)(21,35,36,29)(24,31,40,37)(25,41,42,38)(33,43,46,45)(39,47,48,44);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,20,30,32,22,34,19,28)(18,24,23,37,27,40,26,31)(21,33,35,45,36,46,29,43)(25,39,38,47,42,48,41,44);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,21)(18,25)(19,29)(20,33)(22,36)(23,38)(24,39)(26,41)(27,42)(28,43)(30,35)(31,44)(32,45)(34,46)(37,47)(40,48);(2,5)(6,8)(10,14)(13,15)(20,34)(21,36)(24,40)(25,42)(28,32)(29,35)(31,37)(38,41)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dhi,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7237244);