Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| 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 | #38032 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1535 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)(19,23)(20,31)(21,25)(22,27)(24,34)(26,32)(28,41)(29,36)(30,37)(33,40)(35,43)(38,42)(39,45)(44,46)(47,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,29)(18,23,27,36)(20,28,33,44)(21,30,32,42)(24,35,39,47)(25,37,26,38)(31,41,40,46)(34,43,45,48);(2,6)(3,8)(10,13)(14,16)(17,20,32,45,22,33,21,34)(18,24,25,40,27,39,26,31)(19,28,42,48,29,44,30,43)(23,35,37,46,36,47,38,41);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(19,29)(20,33)(23,36)(24,39)(30,42)(31,40)(34,45)(37,38);(10,13)(14,16)(17,21,22,32)(18,25,27,26)(19,30,29,42)(20,34,33,45)(23,37,36,38)(24,40,39,31)(28,43,44,48)(35,46,47,41)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=fh,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=efh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6693581);