Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3384 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)(11,20)(12,18)(13,19)(14,16)(15,22)(17,23)(21,24);(1,2)(3,4)(5,6)(7,8)(9,11,13,22)(10,15,19,20)(12,23,14,21)(16,17,18,24);(6,8)(10,16)(12,14)(15,17)(18,19)(20,24)(21,23);(2,4)(9,11)(10,17)(12,21)(13,22)(14,23)(15,16)(18,20)(19,24);(9,12,13,14)(10,18,19,16)(11,21,22,23)(15,17,20,24)Polycyclic presentation:
2^9|a^2=fi,b^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72315640, #72315761, #72315788 |
| maximal normal subgroup of | #72315640, #72315761, #72315788 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408176);