Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1595 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(5,8)(11,15)(14,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(1,2,4,6)(3,5,7,8)(10,13)(14,16)(18,22)(19,25)(24,28)(27,31)(36,44)(39,46)(42,48)(45,47);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,26)(20,28)(21,22)(23,25)(24,29)(27,30)(31,32)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(9,11)(10,14)(12,15)(13,16)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=g,b^2=fghi,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5135664);