Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1367 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(9,10,14,12)(11,15,13,16)(17,18,21,24)(19,25,26,22)(20,28,29,23)(27,30,32,31)(33,34)(35,36)(37,42)(38,39)(40,41)(43,44)(45,48)(46,47);(9,11,14,13)(10,15,12,16)(19,26)(20,29)(22,25)(23,28)(34,39)(35,41)(42,47)(43,45);(9,12,14,10)(11,15,13,16)(17,19)(18,22)(20,27)(21,26)(23,30)(24,25)(28,31)(29,32)(33,35)(34,40)(36,39)(37,45)(38,41)(42,44)(43,46)(47,48);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,16,12,15)(17,18)(19,25)(20,23)(21,24)(22,26)(27,31)(28,29)(30,32)(33,36,38,40)(34,41,39,35)(37,44,46,48)(42,45,47,43);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,15,12,16)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)(33,37,38,46)(34,42,39,47)(35,43,41,45)(36,44,40,48)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=gh,d^2=fgh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5463740);