Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,5)(6,8)(11,15)(13,16)(17,18,21,22)(19,25,26,23)(20,24,29,28)(27,31,32,30)(33,34)(35,37)(36,40)(38,42)(39,41)(43,44)(45,48)(46,47);(1,3,5,8)(2,6,4,7)(18,22)(19,26)(24,28)(27,32)(33,35)(34,39)(36,43)(37,42)(38,41)(40,46)(44,48)(45,47);(2,4)(6,7)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,31)(30,32)(33,35)(34,37)(36,43)(38,41)(39,42)(40,44)(45,47)(46,48);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31)(33,36,38,45)(34,40,42,48)(35,43,41,47)(37,44,39,46);(1,4,5,2)(3,7,8,6)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)(33,37)(34,41)(35,42)(36,44)(38,39)(40,47)(43,48)(45,46)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gi,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4544760);