Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C25 | C24 |
| commutator | C24 | C23×C4 |
| prime radical | C24×#41 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
18866 conjugacy classes of subgroups; only the 1010 normal subgroups are shown.
This group has 18866 conjugacy classes of subgroups — too many to draw. Only its 1010 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) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,7,8)(10,12)(14,16)(19,23)(22,24);(9,10)(11,12)(18,21)(22,24);(13,14)(15,16)(17,18)(19,22)(20,21)(23,24);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^9|a^2=i,d^2=g,b^a=be,c^a=cf,d^a=dh,c^b=cg128 conjugacy classes — view the full character table.
| Fitting subgroup of | #72268821 |
| maximal subgroup of | #72237643, #72237693, #72250140, … |
| maximal normal subgroup of | #72237643, #72237693, #72250140, … |
| maximal quotient of | #72237670, #72237712, #72250611, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 61319);