Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(3,7)(6,8)(9,11,14,12)(10,15,13,16)(18,22)(20,29)(23,25)(26,32)(34,36)(37,45)(39,41)(42,48);(1,2,4,5)(3,6,7,8)(9,12,14,11)(10,15,13,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,3,4,7)(2,6,5,8)(9,13,14,10)(11,15,12,16)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,36,38,34)(35,39,43,41)(37,44,45,40)(42,46,48,47);(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=fh,b^2=hi,c^2=g,d^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4384437);