Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628867 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #32888 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,18)(12,16)(13,21)(14,17)(15,19)(20,23)(22,24);(2,5)(6,8);(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(12,13)(16,21)(18,23)(19,22);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)Polycyclic presentation:
2^9|e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,g^c=gi,h^c=hi,e^d=ei,g^d=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239476, #72278281, #72278302 |
| maximal normal subgroup of | #72239476, #72278281, #72278302 |
| maximal quotient of | #72264531, #72264560, #72264646 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536568);