Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
1358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(17,18,22,25)(19,20,26,23)(21,29,30,24)(27,31,32,28)(33,34,37,41)(35,39,43,38)(36,40,45,48)(42,47,44,46);(10,14)(15,16)(19,26)(20,23)(27,32)(28,31)(33,35,37,43)(34,38,41,39)(36,44,45,42)(40,47,48,46);(9,10,13,14)(11,15,12,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,31)(28,29)(30,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,45)(46,48);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,15,14,16)(17,20,22,23)(18,19,25,26)(21,28,30,31)(24,27,29,32)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,47)(45,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,21,22,30)(18,24,25,29)(19,27,26,32)(20,28,23,31)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,48)(43,44)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fgh,d^2=fh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1532832);