Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460131 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1385 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15,20,24)(12,16)(13,21,23,17)(14,18)(19,22);(9,11)(10,15,18,24)(12,21,22,17)(13,19)(14,20)(16,23);(2,5)(6,8)(9,12)(10,16)(11,17)(13,15)(14,22)(18,19)(20,21)(23,24);(1,3,4,7)(2,6,5,8)(9,13)(10,17)(11,19)(12,15)(14,23)(16,20)(18,21)(22,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=f,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,h^c=hi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72225672, #72225682, #72225683, … |
| maximal normal subgroup of | #72225672, #72225682, #72225683, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378195);