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 | |
1666 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)(17,18,22,28)(19,30,33,24)(20,26,35,42)(21,29,23,27)(25,37,41,34)(31,45,47,39)(32,40,36,43)(38,48,44,46);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,33)(18,24,28,30)(20,31,35,47)(21,36,23,32)(25,38,41,44)(26,39,42,45)(27,43,29,40)(34,48,37,46);(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,31)(21,34)(22,35)(23,37)(24,38)(26,27)(28,41)(29,42)(30,44)(32,46)(33,47)(36,48)(39,43)(40,45);(3,8)(5,7)(10,14)(11,16)(17,20)(18,26)(19,31)(21,37)(22,35)(23,34)(24,39)(25,27)(28,42)(29,41)(30,45)(32,48)(33,47)(36,46)(38,43)(40,44);(2,6)(5,7)(10,14)(13,15)(17,21,22,23)(18,27,28,29)(19,32,33,36)(20,34,35,37)(24,40,30,43)(25,26,41,42)(31,46,47,48)(38,45,44,39)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^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,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, 7208521);