Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5179 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)(6,8)(9,10,16,20)(11,21,12,18)(13,14,19,15)(17,23,22,24);(1,2,3,4)(9,11,15,24,16,12,14,23)(10,17,19,21,20,22,13,18);(5,6,7,8)(9,12)(10,18)(11,16)(13,17)(14,23)(15,24)(19,22)(20,21);(9,13)(10,14)(11,22)(12,17)(15,20)(16,19)(18,23)(21,24);(9,14,16,15)(10,19,20,13)(11,23,12,24)(17,21,22,18)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72305325, #72305330, #72305332, … |
| maximal normal subgroup of | #72305325, #72305330, #72305332, … |
| maximal quotient of | #72305504, #72305602, #72305678, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350391);