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 | |
5133 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)(9,10,15,11)(12,16,21,19)(13,18,14,17)(20,22,24,23);(1,2,3,4)(9,11,13,18,15,10,14,17)(12,20,22,19,21,24,23,16);(5,6,7,8)(9,12)(10,16)(11,19)(13,23)(14,22)(15,21)(17,24)(18,20);(9,10,15,11)(12,16,21,19)(13,18,14,17)(20,22,24,23);(12,21)(16,19)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^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, #72305327, #72305329, … |
| maximal normal subgroup of | #72305325, #72305327, #72305329, … |
| maximal quotient of | #72305874, #72305876, #72305878, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350387);