Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479303 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17198 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
874 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 874 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)(3,7,13,19)(4,8)(5,9,16,25)(6,10)(11,20,27,17)(12,23)(14,24,30,21)(15,22)(18,29)(26,32)(28,31);(1,3)(2,7,10,19)(4,11)(5,12)(6,13)(8,20,22,17)(9,23,25,18)(14,26)(15,27)(16,29)(21,31,24,32)(28,30);(1,4)(2,8)(3,11)(5,14)(6,15)(7,17)(9,24)(10,22)(12,28)(13,27)(16,30)(18,31)(19,20)(21,25)(23,32)(26,29);(1,5)(2,9,10,25)(3,12)(4,14)(6,16)(7,18,19,23)(8,21,22,24)(11,26)(13,29)(15,30)(17,31,20,32)(27,28)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,d^2=ef,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,h^c=hi,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72237857, #72237861, #72237862, … |
| maximal normal subgroup of | #72237857, #72237861, #72237862, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386499);