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) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15947 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,12)(11,15,16,14)(17,18)(19,22)(20,24)(21,23)(26,28)(27,31);(4,8)(6,7)(10,12)(11,16)(17,19)(18,22)(20,23)(21,24)(25,26,29,28)(27,32,31,30);(2,3)(6,7)(10,12)(14,15)(18,23)(19,24)(26,28)(27,31);(1,3,5,2)(4,7,8,6)(9,11,13,16)(10,14,12,15)(17,19)(18,20)(21,24)(22,23)(25,27,29,31)(26,30,28,32);(1,4)(2,6)(3,7)(5,8)(9,12,13,10)(11,15,16,14)(17,20)(18,19)(21,22)(23,24)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|a^2=fgi,b^2=gi,d^2=gh,e^2=f,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.
| maximal subgroup of | #72328657, #72328667, #72328723 |
| maximal normal subgroup of | #72328657, #72328667, #72328723 |
| maximal quotient of | #72331381 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4974996);