Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1603 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,12)(13,14)(15,16)(19,25)(20,27)(23,28)(24,29)(33,34)(35,41)(36,38)(37,40)(39,43)(42,48)(44,45)(46,47);(1,3,5,7)(2,4,6,8)(9,10)(11,12)(13,14)(15,16)(18,22)(23,28)(24,29)(30,32)(34,36)(37,45)(39,41)(42,46);(2,6)(3,7)(10,14)(11,15)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,38,43)(34,39,36,41)(37,46,45,42)(40,48,44,47);(1,2)(3,4)(5,6)(7,8)(9,11,13,15)(10,12,14,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,48,46,47);(1,4)(2,3)(5,8)(6,7)(9,12)(10,15)(11,14)(13,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,47)(41,48)(43,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=hi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5030967);