Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1437 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(9,10,13,12)(11,14,16,15)(18,22)(19,25)(20,27)(30,32)(33,34)(35,38)(36,41)(37,40)(39,43)(42,45)(44,48)(46,47);(9,10,13,12)(11,15,16,14)(19,25)(23,28)(26,31)(30,32)(34,35)(36,43)(37,45)(47,48);(1,2,5,4)(3,6,7,8)(9,11,13,16)(10,14,12,15)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,38,34)(36,39,43,41)(37,42,45,40)(44,47,46,48);(1,3)(2,6)(4,8)(5,7)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,36)(34,39)(35,41)(37,46)(38,43)(40,48)(42,47)(44,45);(1,4,5,2)(3,8,7,6)(9,12,13,10)(11,15,16,14)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,47)(41,48)(43,46)Polycyclic presentation:
2^9|a^2=fh,b^2=fh,c^2=fhi,d^2=fg,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3165960);