Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142966 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | #18164 | C2 |
| LCS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
667 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 667 conjugacy classes of subgroups — too many to draw. Only its 56 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:
(1,2,8,23)(3,12,16,31)(4,18,24,29)(5,6,7,11)(9,25,26,14)(10,22,21,20)(13,30,27,19)(15,28,17,32);(1,3)(2,9)(4,19)(5,15)(6,13)(7,17)(8,16)(10,28)(11,27)(12,18)(14,20)(21,32)(22,25)(23,26)(24,30)(29,31);(2,6)(3,13)(4,20)(9,15)(11,23)(12,30)(14,32)(16,27)(17,26)(19,31)(22,24)(25,28)Polycyclic presentation:
2^9|a^2=gi,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,f^b=fg,h^b=hi,d^c=dh,f^c=fg,f^d=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72233828, #72233829, #72233830, … |
| maximal normal subgroup of | #72233828, #72233829, #72233830, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50176);