Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(9,10,13,12)(11,14,15,16)(17,18)(19,22)(20,25)(21,26)(23,28)(24,30)(27,31)(29,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,48)(44,46)(45,47);(11,15)(14,16)(18,19)(23,25)(24,26)(31,32)(33,34,37,40)(35,41,43,38)(36,39,46,44)(42,48,45,47);(9,11)(10,14)(12,16)(13,15)(17,19,22,18)(20,25,28,23)(21,24,30,26)(27,31,29,32)(33,34)(35,38)(36,44)(37,40)(39,46)(41,43)(42,48)(45,47);(1,2)(3,5)(4,6)(7,8)(17,20)(18,23)(19,25)(21,29)(22,28)(24,32)(26,31)(27,30)(33,35,37,43)(34,38,40,41)(36,45,46,42)(39,48,44,47);(1,3,4,7)(2,5,6,8)(9,12,13,10)(11,16,15,14)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)(33,36,37,46)(34,39,40,44)(35,42,43,45)(38,47,41,48)Polycyclic presentation:
2^9|a^2=hi,b^2=fg,c^2=g,d^2=fg,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3553731);