Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,3)(4,6,8,7)(9,10,13,16)(11,14,12,15)(18,22)(19,25)(20,28)(30,32)(33,34,38,36)(35,39,41,43)(37,44,46,40)(42,48,45,47);(4,8)(6,7)(9,11,13,12)(10,14,16,15)(18,22)(20,28)(23,26)(27,31)(34,36)(35,41)(40,44)(42,45);(2,3)(6,7)(9,11,13,12)(10,14,16,15)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,35)(34,39)(36,43)(37,45)(38,41)(40,48)(42,46)(44,47);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,36,38,34)(35,39,41,43)(37,44,46,40)(42,47,45,48);(1,4)(2,6)(3,7)(5,8)(9,12,13,11)(10,15,16,14)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=fhi,c^2=i,d^2=gh,e^2=fhi,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, 5116798);