Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #25762 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1719 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)(3,6)(4,5)(7,8)(10,13)(11,15)(17,18,23,29,24,30,22,28)(19,25,32,41,33,42,38,45)(20,34,36,44,21,27,37,26)(31,40,47,43,35,48,46,39);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,24,33)(18,25,30,42)(20,35,21,31)(22,38,23,32)(26,43,44,39)(27,40,34,48)(28,45,29,41)(36,46,37,47);(10,13)(14,16)(17,20,24,21)(18,26,30,44)(19,31,33,35)(22,36,23,37)(25,39,42,43)(27,28,34,29)(32,46,38,47)(40,41,48,45);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,21,24,20)(18,27,30,34)(19,31,33,35)(22,36,23,37)(25,40,42,48)(26,28,44,29)(32,47,38,46)(39,41,43,45);(1,3,4,7)(2,6,5,8)(17,22,24,23)(18,28,30,29)(19,32,33,38)(20,36,21,37)(25,41,42,45)(26,34,44,27)(31,46,35,47)(39,48,43,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=h,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,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, 6778801);