Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| 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 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5151 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)(9,10,14,17)(11,18,16,13)(12,15,20,22)(19,24,23,21);(5,6)(7,8)(9,11,13,10,14,16,18,17)(12,19,21,15,20,23,24,22);(1,2,3,4)(6,8)(9,12)(10,15)(11,19)(13,21)(14,20)(16,23)(17,22)(18,24);(6,8);(12,20)(15,22)(19,23)(21,24)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72305327, #72305332, #72305340, … |
| maximal normal subgroup of | #72305327, #72305332, #72305340, … |
| maximal quotient of | #72305764, #72305912, #72306169, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350305);