Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499867 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | #62817 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4882 conjugacy classes of subgroups; only the 590 normal subgroups are shown.
This group has 4882 conjugacy classes of subgroups — too many to draw. Only its 590 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24);(1,3)(2,5)(4,7)(6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23);(12,20)(15,24)(17,19)(21,23);(11,18)(14,22)(17,19)(21,23)Polycyclic presentation:
2^9|a^2=h,b^a=bf,c^a=cg,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi80 conjugacy classes — view the full character table.
| Fitting subgroup of | #72267337 |
| maximal subgroup of | #72236483, #72240169, #72240170, … |
| maximal normal subgroup of | #72236483, #72240169, #72240170, … |
| maximal quotient of | #72236769, #72251070, #72263494, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6415888);