Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500535 |
| 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 | #25131 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1664 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)(3,6)(4,5)(7,8)(17,18)(19,28)(20,25)(21,26)(22,37)(23,27)(24,31)(29,40)(30,45)(32,48)(33,42)(34,44)(35,38)(36,43)(39,47)(41,46);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(19,29)(24,38)(28,40)(30,32)(31,35)(39,41)(45,48)(46,47);(9,11)(10,13)(12,16)(14,15)(17,19,22,31,23,29,21,35)(18,24,26,28,27,38,37,40)(20,32,33,46,34,30,36,47)(25,41,43,48,44,39,42,45);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20,23,34)(18,25,27,44)(19,30,29,32)(21,36,22,33)(24,39,38,41)(26,43,37,42)(28,45,40,48)(31,47,35,46);(10,14)(13,15)(17,21,23,22)(18,26,27,37)(19,31,29,35)(20,33,34,36)(24,40,38,28)(25,42,44,43)(30,46,32,47)(39,45,41,48)Polycyclic presentation:
2^9|c^2=g,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6414176);