Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5587 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10,13,17,14,18,12,16)(11,19,20,23,21,24,22,15);(5,6)(7,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,19)(17,24)(18,23);(2,4)(6,8);(6,8);(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,19,23,24)Polycyclic presentation:
2^9|a^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72305327, #72305328, #72305339, … |
| maximal normal subgroup of | #72305327, #72305328, #72305339, … |
| maximal quotient of | #72306139, #72306153, #72306226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344927);