Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628842 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36614 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
17825 conjugacy classes of subgroups; only the 1018 normal subgroups are shown.
This group has 17825 conjugacy classes of subgroups — too many to draw. Only its 1018 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) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,22)(19,24)(21,23);(9,11,13,19)(10,14,15,23)(12,21,22,18)(16,24,20,17);(2,4);(6,8);(10,15)(14,23)(16,20)(17,24)Polycyclic presentation:
2^9|b^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fi,f^b=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72258260, #72258264, #72258278, … |
| maximal normal subgroup of | #72258260, #72258264, #72258278, … |
| maximal quotient of | #72306224, #72306229, #72307566, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536039);