Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | #28146 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,28,22,27,38,26)(19,29,31,42,32,23,39,43)(20,33,37,45,36,25,35,24)(30,41,48,40,34,46,47,44);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,32)(18,23,27,29)(20,34,36,30)(21,31,38,39)(24,44,45,40)(25,46,33,41)(26,42,28,43)(35,48,37,47);(9,11)(10,13)(12,16)(14,15)(17,20,22,36)(18,24,27,45)(19,30,32,34)(21,35,38,37)(23,40,29,44)(25,26,33,28)(31,47,39,48)(41,42,46,43);(3,8)(6,7)(11,16)(13,15)(17,20)(18,25)(19,30)(21,37)(22,36)(23,41)(24,28)(26,45)(27,33)(29,46)(31,48)(32,34)(35,38)(39,47)(40,43)(42,44);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,21,22,38)(18,26,27,28)(19,31,32,39)(20,35,36,37)(23,42,29,43)(24,33,45,25)(30,47,34,48)(40,46,44,41)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6919431);