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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1327 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)(17,18,23,27,22,28,21,37)(19,29,33,46,32,24,31,41)(20,26,35,44,36,43,38,25)(30,45,47,39,34,40,48,42);(1,2,5,6)(3,8,4,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,32)(18,24,28,29)(20,34,36,30)(21,31,23,33)(25,42,44,39)(26,45,43,40)(27,41,37,46)(35,48,38,47);(9,11)(10,13)(12,16)(14,15)(17,20,22,36)(18,25,28,44)(19,30,32,34)(21,35,23,38)(24,39,29,42)(26,27,43,37)(31,47,33,48)(40,41,45,46);(1,3,5,4)(2,7,6,8)(11,16)(13,15)(17,20)(18,26)(19,30)(21,38)(22,36)(23,35)(24,40)(25,37)(27,44)(28,43)(29,45)(31,48)(32,34)(33,47)(39,46)(41,42);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,22,23)(18,27,28,37)(19,31,32,33)(20,35,36,38)(24,41,29,46)(25,43,44,26)(30,47,34,48)(39,45,42,40)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,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, 6922745);