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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1306 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)(9,10,13,11)(12,14,16,15)(18,22)(19,25)(24,29)(27,31)(34,35)(36,43)(37,45)(46,47);(10,11)(12,16)(20,28)(24,29)(27,31)(30,32)(33,34,38,35)(36,39,43,41)(37,40,45,42)(44,46,48,47);(9,11,13,10)(12,14,16,15)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,35,38,34)(36,39,43,41)(37,40,45,42)(44,47,48,46);(1,2)(3,5)(4,6)(7,8)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,36)(34,39)(35,41)(37,44)(38,43)(40,46)(42,47)(45,48);(1,3,4,7)(2,5,6,8)(9,12)(10,14)(11,15)(13,16)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,37,38,45)(34,40,35,42)(36,44,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=fh,c^2=gh,d^2=fgh,e^2=fghi,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, 5512861);