Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2613 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6093 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,21,22)(19,24,23,20);(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,23)(21,22);(6,7)(10,13)(14,15)(18,22)(19,23);(1,2,3,4)(5,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,19,21,23)(18,20,22,24);(9,11)(10,14)(12,16)(13,15)(17,20)(18,23)(19,22)(21,24)Polycyclic presentation:
2^9|a^2=f,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
| maximal subgroup of | #72219257, #72226011, #72234112, … |
| maximal normal subgroup of | #72219257, #72219259, #72226011, … |
| maximal quotient of | #72332493, #72332601, #72333296 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4837691);