Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,3)(4,8)(10,14)(12,16)(18,22)(19,25)(20,27)(30,32)(33,34,37,40)(35,38,41,45)(36,43,44,39)(42,48,47,46);(1,2,5,3)(4,7,8,6)(9,10)(11,12)(13,14)(15,16)(18,22)(19,25)(24,29)(26,31)(35,41)(38,45)(42,47)(46,48);(2,3)(6,7)(10,14)(11,15)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,48)(44,47);(1,3,5,2)(4,7,8,6)(9,11,13,15)(10,12,14,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48);(1,4)(2,6)(3,7)(5,8)(9,12)(10,15)(11,14)(13,16)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48)Polycyclic presentation:
2^9|a^2=fgh,b^2=fhi,c^2=g,d^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4384449);