Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #407011 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14945 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1159 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 1159 conjugacy classes of subgroups — too many to draw. Only its 208 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:
(5,17)(7,18)(8,20)(10,22)(12,25)(13,27)(16,28)(19,29);(1,2)(3,11,14,24)(4,8)(5,10)(6,9)(7,13,18,27)(12,23,25,31)(15,20)(16,21)(17,22)(19,26,29,32)(28,30);(1,3)(2,7,9,18)(4,12)(5,13)(6,14)(8,19,20,29)(10,11,22,24)(15,25)(16,26)(17,27)(21,23,30,31)(28,32);(1,4)(2,8)(3,12)(5,16)(6,15)(7,19)(9,20)(10,21)(11,23)(13,26)(14,25)(17,28)(18,29)(22,30)(24,31)(27,32)Polycyclic presentation:
2^9|b^2=efh,c^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=fi,h^b=hi,h^c=hi,e^d=ei,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72338494, #72338495, #72338498 |
| maximal normal subgroup of | #72338494, #72338495, #72338498 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 314207);