Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245225 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #35512 | C2 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
2268 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 2268 conjugacy classes of subgroups — too many to draw. Only its 156 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 — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,18)(11,15,13,17)(12,16,21,24)(19,23,20,22);(1,3)(2,5)(4,7)(6,8)(9,11,14,13)(10,15,18,17)(12,19,21,20)(16,22,24,23);(5,8)(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24);(9,13,14,11)(10,17,18,15)(12,20,21,19)(16,23,24,22)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=h,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,f^d=fh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72236720, #72240639, #72240640, … |
| maximal normal subgroup of | #72236720, #72240639, #72240640, … |
| maximal quotient of | #72236742, #72247306, #72265994, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152426);