Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| 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 | |
1372 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)(11,12)(15,16)(17,18,22,20)(19,25,26,23)(21,28,29,24)(27,30,32,31)(33,34,38,39)(35,41,42,36)(37,45,46,40)(43,44,48,47);(9,10,14,13)(11,15,12,16)(19,26)(23,25)(27,32)(30,31)(34,39)(35,42)(37,46)(44,47);(9,11,14,12)(10,15,13,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32)(33,35,38,42)(34,36,39,41)(37,43,46,48)(40,44,45,47);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,16,13,15)(17,20,22,18)(19,25,26,23)(21,28,29,24)(27,31,32,30)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,48)(46,47);(1,3,4,7)(2,6,5,8)(9,13,14,10)(11,16,12,15)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37,38,46)(34,40,39,45)(35,43,42,48)(36,44,41,47)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=fgh,d^2=fg,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4272874);