Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1735 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)(19,24)(20,32)(21,26)(22,37)(23,27)(25,35)(28,44)(29,39)(30,40)(31,47)(33,48)(34,42)(36,43)(38,46)(41,45);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,29)(18,24,27,39)(20,33,36,28)(21,30,22,31)(25,41,42,38)(26,40,37,47)(32,48,43,44)(34,46,35,45);(10,13)(14,16)(17,20,23,36)(18,25,27,42)(19,28,29,33)(21,35,22,34)(24,38,39,41)(26,43,37,32)(30,46,31,45)(40,48,47,44);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(19,29)(24,39)(28,33)(30,31)(38,41)(40,47)(44,48)(45,46);(3,8)(5,7)(10,13)(14,16)(17,21,23,22)(18,26,27,37)(19,30,29,31)(20,34,36,35)(24,40,39,47)(25,32,42,43)(28,45,33,46)(38,44,41,48)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,e^c=ei,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6675151);