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 | #42760 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1347 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,25,21,27,22,28)(19,29,36,40,31,24,32,38)(20,26,37,41,34,43,35,44)(30,42,46,48,33,39,45,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19,21,31)(18,24,27,29)(20,33,34,30)(22,32,23,36)(25,38,28,40)(26,42,43,39)(35,46,37,45)(41,48,44,47);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(18,25)(22,23)(24,38)(26,41)(27,28)(29,40)(32,36)(35,37)(39,47)(42,48)(43,44)(45,46);(1,3,5,4)(2,8,6,7)(11,16)(13,15)(17,20,21,34)(18,26,27,43)(19,30,31,33)(22,35,23,37)(24,39,29,42)(25,41,28,44)(32,45,36,46)(38,47,40,48);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(20,34)(26,43)(30,33)(35,37)(39,42)(41,44)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=fi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,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, 6923728);