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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,4)(5,8)(9,10,13,11)(12,15,16,14)(18,22)(20,28)(23,25)(26,32)(33,34)(35,41)(36,44)(37,38)(39,43)(40,45)(42,46)(47,48);(1,2,6,4)(3,5,7,8)(18,22)(20,28)(23,25)(26,32)(34,37)(39,41)(40,44)(46,47);(1,3,6,7)(2,5,4,8)(9,11,13,10)(12,15,16,14)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,47)(45,48);(1,4,6,2)(3,5,7,8)(9,12,13,16)(10,14,11,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36,38,45)(34,40,37,44)(35,42,43,48)(39,46,41,47);(1,5)(2,7)(3,4)(6,8)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,34)(35,41,43,39)(36,40,45,44)(42,46,48,47)Polycyclic presentation:
2^9|a^2=fi,b^2=fgh,c^2=gi,d^2=hi,e^2=gh,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, 5791526);