Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682380 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1408 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)(17,18,22,27)(19,29,32,24)(20,33,35,40)(21,28,23,26)(25,34,41,36)(30,44,46,47)(31,39,37,42)(38,45,43,48);(1,3)(2,5)(4,8)(6,7)(9,10,13,16)(11,15,12,14)(17,19,22,32)(18,24,27,29)(20,30,35,46)(21,37,23,31)(25,38,41,43)(26,42,28,39)(33,47,40,44)(34,48,36,45);(2,6)(3,8)(9,11,13,12)(10,14,16,15)(17,20,21,34,22,35,23,36)(18,25,26,40,27,41,28,33)(19,30,37,48,32,46,31,45)(24,38,42,44,29,43,39,47);(2,6)(5,7)(17,21,22,23)(18,26,27,28)(19,31,32,37)(20,34,35,36)(24,39,29,42)(25,40,41,33)(30,45,46,48)(38,47,43,44);(9,12,13,11)(10,15,16,14)(19,32)(24,29)(30,46)(31,37)(38,43)(39,42)(44,47)(45,48)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=ghi,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6761304);