Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200042 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1782 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)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,23,31,40)(20,25,32,43)(21,34,36,26)(24,35,41,33)(28,42,29,38)(30,46,47,39)(37,45,48,44);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,28)(21,30)(22,31)(24,37)(25,42)(26,39)(27,40)(29,32)(33,44)(34,46)(35,45)(36,47)(38,43)(41,48);(11,15)(13,16)(17,20)(18,24)(19,28)(21,35)(22,32)(23,37)(25,34)(26,43)(27,41)(29,31)(30,45)(33,36)(38,39)(40,48)(42,46)(44,47);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,20)(18,25)(19,29)(21,33)(22,32)(23,38)(24,34)(26,41)(27,43)(28,31)(30,45)(35,36)(37,39)(40,42)(44,47)(46,48);(2,5)(6,8)(11,15)(13,16)(20,32)(24,41)(25,43)(28,29)(33,35)(37,48)(38,42)(44,45)Polycyclic presentation:
2^9|a^2=fi,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,e^c=ei,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7109562);