Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245246 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #36714 | C2 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1828 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 1828 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,9)(5,8)(6,11)(7,14)(10,18)(13,25)(15,27)(16,23)(17,22)(19,32)(20,28)(21,30)(24,31)(26,29);(4,16)(7,19)(9,23)(12,25)(14,30)(17,31)(20,21)(22,24)(26,27)(28,32);(1,3,6,15)(2,7,11,21)(4,14,16,30)(5,13,18,29)(8,20,10,19)(9,12,23,27)(17,28,31,32)(22,25,24,26);(2,8)(4,17)(7,20)(10,11)(14,28)(16,31)(19,21)(30,32)Polycyclic presentation:
2^9|c^2=i,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 | #72234113, #72240629, #72240632, … |
| maximal normal subgroup of | #72234113, #72240629, #72240632, … |
| maximal quotient of | #72251442, #72251581, #72303859 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152454);