Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #436465 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15561 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23445 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1245 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 1245 conjugacy classes of subgroups — too many to draw. Only its 180 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)(3,11,13,26)(4,16,19,9)(5,10)(6,8)(7,15,23,20)(12,21)(14,29)(17,22)(18,30,27,25)(24,28)(31,32);(1,3,12,15,6,13,5,20)(2,7,21,11,8,23,10,26)(4,17,27,28,19,31,18,14)(9,24,30,32,16,29,25,22);(2,8)(3,13)(5,12)(9,16)(11,26)(14,28)(18,27)(24,29);(1,4,6,19)(2,9,8,16)(3,14)(5,18,12,27)(7,22)(10,25,21,30)(11,24)(13,28)(15,17)(20,31)(23,32)(26,29)Polycyclic presentation:
2^9|a^2=g,b^2=ef,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72283686, #72283941, #72285336, … |
| maximal normal subgroup of | #72283686, #72283941, #72285336, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343661);