Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152179 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12759 |
| commutator | #124 | C22×C4 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | #61109 | C2 |
| LCS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
834 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 834 conjugacy classes of subgroups — too many to draw. Only its 72 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 — 3 generators on 32 points:
(1,2)(3,11,14,28)(4,16,17,31)(5,19,20,9)(6,10)(7,12,22,29)(8,18)(13,27,15,24)(21,26,23,30)(25,32);(1,3)(2,7)(4,12)(5,13)(6,15)(8,24)(9,22)(10,23)(11,25)(14,20)(16,27)(17,30)(18,29)(19,21)(26,32)(28,31);(1,2)(3,7)(4,16)(5,19)(6,10)(8,18)(9,20)(11,26)(12,27)(13,21)(14,22)(15,23)(17,31)(24,29)(25,32)(28,30)Polycyclic presentation:
2^9|a^2=f,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dh,e^b=egh,f^b=fghi,g^b=gi,d^c=dg,g^c=gi,h^c=hi,e^d=ehi,f^d=fhi,g^e=gi,g^f=gi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72213674, #72219527, #72219528, … |
| maximal normal subgroup of | #72213674, #72219527, #72219528, … |
| maximal quotient of | #72338281 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59375);