Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #407054 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14946 |
| 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 | |
1332 conjugacy classes of subgroups; only the 224 normal subgroups are shown.
This group has 1332 conjugacy classes of subgroups — too many to draw. Only its 224 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)(4,8,15,22)(5,16,17,9)(6,10)(11,23,25,18)(12,19,26,28)(13,20)(14,21)(24,30)(27,29)(31,32);(1,3)(2,7)(4,11,15,25)(5,12)(6,13)(8,18,22,23)(9,19)(10,20)(14,24,27,32)(16,28)(17,26)(21,30,29,31);(1,4)(2,8,10,22)(3,11,13,25)(5,14)(6,15)(7,18)(9,21,16,29)(12,24,26,32)(17,27)(19,30)(20,23)(28,31);(1,5)(2,9)(3,12)(4,14)(6,17)(7,19)(8,21)(10,16)(11,24)(13,26)(15,27)(18,30)(20,28)(22,29)(23,31)(25,32)Polycyclic presentation:
2^9|a^2=e,b^2=efh,c^2=ef,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237790, #72237801, #72240450, … |
| maximal normal subgroup of | #72237790, #72237801, #72240450, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 314250);