Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #186828 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62825 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2122 conjugacy classes of subgroups; only the 310 normal subgroups are shown.
This group has 2122 conjugacy classes of subgroups — too many to draw. Only its 310 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:
(2,7,9,12)(8,15,16,14)(17,18)(19,25)(20,22)(21,29)(23,27)(24,31)(26,30)(28,32);(1,2)(3,8)(4,7)(5,12)(6,9)(10,14)(11,16)(13,15)(17,19)(18,23)(20,26)(21,30)(22,28)(24,32)(25,29)(27,31);(1,3)(2,8)(4,10)(5,13)(6,11)(7,14)(9,16)(12,15)(17,20,22,18)(19,26,28,23)(21,24,31,29)(25,30,32,27);(1,4,6,5)(2,7,9,12)(3,10,11,13)(8,14,16,15)(18,20)(23,26)(24,29)(25,32)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72239238 |
| maximal normal subgroup of | #72239238 |
| maximal quotient of | #72239246 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94030);