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 | |
3557 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)(9,10,13,15,14,17,11,18)(12,20,19,16,22,21,23,24);(5,6)(7,8)(9,10)(11,15)(12,21)(13,18)(14,17)(16,19)(20,22)(23,24);(2,4)(6,8)(9,11,14,13)(10,15,17,18)(12,19,22,23)(16,20,24,21);(2,4)(9,12,14,22)(10,16,17,24)(11,19,13,23)(15,20,18,21);(10,17)(15,18)(16,24)(20,21)Polycyclic presentation:
2^9|a^2=f,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 | #72281515, #72282093, #72282168, … |
| maximal normal subgroup of | #72281515, #72282093, #72282168, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408013);