Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,4,7,8)(11,16)(12,14)(18,22)(23,25)(24,29)(30,31)(33,34,38,39)(35,36,41,40)(37,42,45,47)(43,44,48,46);(1,2,5,6)(3,4,7,8)(9,10,13,15)(11,12,16,14)(20,28)(24,29)(26,32)(30,31)(34,39)(35,41)(37,45)(44,46);(2,6)(3,7)(9,11,13,16)(10,14,15,12)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,35,38,41)(34,40,39,36)(37,43,45,48)(42,46,47,44);(1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36,38,40)(34,41,39,35)(37,46,45,44)(42,43,47,48);(1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,37)(34,42)(35,43)(36,44)(38,45)(39,47)(40,46)(41,48)Polycyclic presentation:
2^9|a^2=hi,b^2=fh,c^2=fghi,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5637937);