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)(11,17)(12,20,21,15)(13,16)(14,19)(18,22,24,23);(9,11)(10,14,16,17)(12,18)(13,19)(15,22,20,23)(21,24);(2,5)(6,8)(18,24)(22,23);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=f,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,d^c=de,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72225672, #72225683, #72225685, … |
| maximal normal subgroup of | #72225672, #72225683, #72225685, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378192);