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 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(4,8)(9,10,13,11)(12,15,16,14)(18,22)(19,25)(20,28)(30,32)(33,34,37,38)(35,41,43,39)(36,40,45,44)(42,47,48,46);(1,2,5,6)(3,4,7,8)(18,22)(20,28)(23,26)(27,31)(34,38)(39,41)(40,44)(46,47);(1,3,5,7)(2,4,6,8)(9,11,13,10)(12,15,16,14)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48);(1,3)(2,4)(5,7)(6,8)(9,12,13,16)(10,14,11,15)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,36,37,45)(34,40,38,44)(35,42,43,48)(39,46,41,47);(1,4)(2,7)(3,6)(5,8)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,34)(35,39)(36,44)(37,38)(40,45)(41,43)(42,47)(46,48)Polycyclic presentation:
2^9|a^2=gi,b^2=fgh,c^2=fgi,d^2=ghi,e^2=h,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, 5791305);