Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,6,7,8)(9,10,13,11)(12,15,16,14)(17,18,22,23)(19,20,26,24)(21,29,30,25)(27,32,31,28)(33,34)(35,41)(36,39)(37,40)(38,43)(42,48)(44,46)(45,47);(3,7)(6,8)(18,23)(19,26)(21,30)(28,32)(33,34,37,40)(35,38,43,41)(36,39,46,44)(42,47,45,48);(1,3,5,7)(2,6,4,8)(9,11,13,10)(12,15,16,14)(17,19,22,26)(18,24,23,20)(21,31,30,27)(25,28,29,32)(33,34)(35,38)(36,44)(37,40)(39,46)(41,43)(42,48)(45,47);(9,12)(10,14)(11,15)(13,16)(17,20,22,24)(18,19,23,26)(21,32,30,28)(25,31,29,27)(33,35,37,43)(34,38,40,41)(36,45,46,42)(39,48,44,47);(1,4,5,2)(3,8,7,6)(17,21,22,30)(18,25,23,29)(19,27,26,31)(20,28,24,32)(33,36)(34,39)(35,42)(37,46)(38,47)(40,44)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=fg,c^2=ghi,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4134547);