Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #407013 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14945 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1260 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 1260 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:
(1,2)(3,7)(4,8)(5,9)(6,10)(11,20)(12,26)(13,19)(14,28)(15,22)(16,23)(17,25)(18,27)(21,29)(24,31)(30,32);(1,3)(2,7)(4,11,15,25)(5,12)(6,13)(8,20,22,17)(9,18)(10,19)(14,24,29,32)(16,27)(21,31,28,30)(23,26);(1,4)(2,8)(3,11,13,25)(5,14)(6,15)(7,17,19,20)(9,21)(10,22)(12,24,27,32)(16,29)(18,30,26,31)(23,28);(1,5)(2,9)(3,12)(4,14)(6,16)(7,18)(8,21)(10,23)(11,24)(13,27)(15,29)(17,30)(19,26)(20,31)(22,28)(25,32)Polycyclic presentation:
2^9|b^2=efh,c^2=ef,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237786, #72237801, #72237893, … |
| maximal normal subgroup of | #72237786, #72237801, #72237893, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 314209);