Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×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×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3363 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)(3,4)(9,10,15,24,14,19,12,17)(11,20,18,16,21,23,22,13);(5,6)(7,8)(9,11)(10,16)(12,18)(13,19)(14,21)(15,22)(17,23)(20,24);(2,4)(6,8)(9,12,14,15)(10,17,19,24)(11,18,21,22)(13,20,16,23);(2,4)(9,13,14,16)(10,18,19,22)(11,17,21,24)(12,20,15,23);(10,19)(11,21)(17,24)(18,22)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,f^b=fi,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72315707, #72315774, #72315989 |
| maximal normal subgroup of | #72315707, #72315774, #72315989 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408014);