Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(10,13)(11,15)(17,18,21,24)(19,25,26,22)(20,23,28,30)(27,31,32,29)(33,34,38,39)(35,42,41,37)(36,40,44,47)(43,45,48,46);(1,2,5,4)(3,6,7,8)(10,13)(14,16)(19,26)(22,25)(27,32)(29,31)(34,39)(35,41)(36,44)(45,46);(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)(33,35,38,41)(34,37,39,42)(36,43,44,48)(40,46,47,45);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,26)(18,22,24,25)(20,27,28,32)(23,29,30,31)(33,36)(34,40)(35,43)(37,46)(38,44)(39,47)(41,48)(42,45);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)(17,20,21,28)(18,23,24,30)(19,27,26,32)(22,29,25,31)(33,37)(34,41)(35,39)(36,45)(38,42)(40,43)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgi,c^2=f,d^2=gh,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4825838);