Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245246 |
| 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 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
2311 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 2311 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)(9,10)(11,14)(12,15)(13,17)(16,19)(18,22)(20,23)(21,24);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(5,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24);(9,11)(10,16)(12,18)(13,19)(14,17)(15,22)(20,24)(21,23)Polycyclic presentation:
2^9|b^2=hi,b^a=be,c^a=cf,d^a=deh,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,f^d=fh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72236720, #72237496, #72240624, … |
| maximal normal subgroup of | #72236720, #72237496, #72240624, … |
| maximal quotient of | #72236728, #72237499, #72240175, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152450);