Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300980 |
| 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 | #42908 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1668 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)(3,7)(4,6)(5,8)(17,18,21,26)(19,28,30,23)(20,25,32,41)(22,35,34,27)(24,36,40,33)(29,43,45,38)(31,39,47,44)(37,46,42,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,30)(18,23,26,28)(20,29,32,45)(22,31,34,47)(24,37,40,42)(25,38,41,43)(27,39,35,44)(33,46,36,48);(9,11)(10,13)(12,16)(14,15)(17,20)(18,24)(19,29)(21,32)(22,36)(23,37)(25,35)(26,40)(27,41)(28,42)(30,45)(31,48)(33,34)(38,44)(39,43)(46,47);(3,8)(5,7)(10,14)(11,16)(17,20)(18,25)(19,29)(21,32)(22,33)(23,38)(24,35)(26,41)(27,40)(28,43)(30,45)(31,46)(34,36)(37,44)(39,42)(47,48);(2,6)(5,7)(10,14)(13,15)(18,26)(19,30)(20,32)(27,35)(31,47)(33,36)(37,42)(38,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,d^c=dh,e^c=ei,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7208584);