Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1302 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)(9,10,13,15)(11,12,14,16)(17,18)(19,20)(21,29)(22,23)(24,26)(25,32)(27,31)(28,30)(33,34)(35,39)(36,42)(37,45)(38,41)(40,46)(43,48)(44,47);(9,11)(10,12)(13,14)(15,16)(18,23)(19,26)(25,29)(27,30)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(9,12)(10,14)(11,15)(13,16)(17,19)(18,24)(20,23)(21,30)(22,26)(25,28)(27,32)(29,31)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,2)(3,6)(4,5)(7,8)(17,20,22,24)(18,19,23,26)(21,31,32,28)(25,30,29,27)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,47)(45,46);(1,3,4,7)(2,6,5,8)(17,21,22,32)(18,25,23,29)(19,27,26,30)(20,28,24,31)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=fg,d^2=f,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3504328);