Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(17,18)(19,29)(20,26)(21,27)(22,28)(23,38)(24,33)(25,35)(30,40)(31,45)(32,46)(34,41)(36,43)(37,44)(39,48)(42,47);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,31)(21,32)(22,33)(23,34)(25,42)(26,40)(27,41)(28,29)(30,36)(35,48)(37,47)(38,46)(39,44)(43,45);(9,11)(10,13)(12,16)(14,15)(17,20,22,36)(18,25,28,44)(19,30,33,31)(21,35,23,37)(24,39,29,42)(26,27,43,38)(32,47,34,48)(40,46,45,41);(11,16)(13,15)(17,20)(18,26)(19,31)(21,37)(22,36)(23,35)(24,40)(25,38)(27,44)(28,43)(29,45)(30,33)(32,47)(34,48)(39,41)(42,46);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,22,23)(18,27,28,38)(19,32,33,34)(20,35,36,37)(24,41,29,46)(25,43,44,26)(30,47,31,48)(39,40,42,45)Polycyclic presentation:
2^9|a^2=fi,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6409438);