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 | #32898 | C2 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
2242 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 2242 conjugacy classes of subgroups — too many to draw. Only its 152 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)(14,20)(17,23);(1,3)(2,5)(4,7)(6,8)(9,10,13,16)(11,14,18,20)(12,15,19,22)(17,21,23,24);(5,8)(9,11)(10,14)(12,17)(13,18)(15,21)(16,20)(19,23)(22,24);(9,12)(10,15)(11,17)(13,19)(14,21)(16,22)(18,23)(20,24)Polycyclic presentation:
2^9|b^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 | #72237496, #72240626, #72240629, … |
| maximal normal subgroup of | #72237496, #72240626, #72240629, … |
| maximal quotient of | #72237497, #72240170, #72247306, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152423);