Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483956 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | #33748 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
475 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 475 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,3,10,8)(4,11,12,15,13,7,14,16)(17,18,19,23,21,26,27,25)(20,29,28,24,31,22,30,32);(1,3)(2,6)(4,11)(5,8)(7,13)(9,10)(12,15)(14,16)(18,22)(19,27)(20,30)(23,24)(25,32)(26,29)(28,31);(1,4,10,14,6,13,5,12)(2,7,8,15,3,11,9,16)(18,23,26,25)(20,31)(22,24,29,32)(28,30);(2,3)(7,11)(8,9)(15,16)(17,19,21,27)(18,23,26,25)(20,28,31,30)(22,32,29,24)Polycyclic presentation:
2^9|a^2=f,c^2=fgi,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dhi,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fi,g^b=gi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213975, #72214568, #72214589, … |
| maximal normal subgroup of | #72213975, #72214568, #72214589, … |
| maximal quotient of | #72216174, C2×#483961, #72225053, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391157);