Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245225 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1580 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 1580 conjugacy classes of subgroups — too many to draw. Only its 156 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8)(3,6,10,12)(4,7,11,14)(9,15,16,13)(17,18)(19,22)(20,23)(21,24)(25,30)(26,29)(27,31)(28,32);(6,12)(13,15)(17,19,21,26)(18,22,24,29)(20,25,27,32)(23,30,31,28);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,16)(14,15)(17,20,21,27)(18,23,24,31)(19,25,26,32)(22,28,29,30);(1,4,5,11)(2,7,8,14)(3,9,10,16)(6,13,12,15)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=h,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,f^d=fh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72341034, #72341037, #72341039, … |
| maximal normal subgroup of | #72341034, #72341037, #72341039, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152437);