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 | |
1302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,4,7)(9,10,13,11)(12,14,16,15)(18,22)(19,25)(20,27)(30,32)(33,34,38,39)(35,36,41,40)(37,45,48,42)(43,47,46,44);(9,11,13,10)(12,14,16,15)(19,25)(23,28)(26,31)(30,32)(34,39)(35,41)(42,45)(43,46);(1,3,5,4)(2,6,8,7)(9,12)(10,14)(11,15)(13,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,38,41)(34,40,39,36)(37,46,48,43)(42,44,45,47);(1,2)(3,6)(4,7)(5,8)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36,38,40)(34,41,39,35)(37,47,48,44)(42,43,45,46);(1,4,5,3)(2,7,8,6)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,37)(34,42)(35,43)(36,44)(38,48)(39,45)(40,47)(41,46)Polycyclic presentation:
2^9|a^2=fghi,b^2=fh,c^2=fi,d^2=f,e^2=fgi,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, 3605032);