Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2433 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(6,8)(10,13)(11,15)(17,18)(19,23)(20,22)(21,24);(2,3)(9,10,12,13)(11,16,15,14)(17,19,22,21)(18,23,20,24);(2,3)(11,15)(14,16)(17,18,22,20)(19,24,21,23);(2,3)(5,6)(7,8)(9,10)(11,16)(12,13)(14,15)(17,20,22,18)(19,23,21,24);(1,3,4,2)(9,11,12,15)(10,14,13,16)(17,21,22,19)(18,23,20,24)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=h,d^2=h,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72234748, #72234813, #72234814, … |
| maximal normal subgroup of | #72234748, #72234813, #72234814, … |
| maximal quotient of | #72298276, #72330016, #72330941, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3781591);