Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142600 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | #3496 | C22 |
| LCS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
577 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 577 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,10)(3,11,13,15)(4,7,5,9)(8,14,16,12)(17,18,22,20)(19,25,28,32)(21,24,30,29)(23,31,26,27);(2,7)(4,5)(8,11)(9,10)(12,14)(15,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,25)(29,32)(30,31);(1,3,6,13)(2,8,10,16)(4,12,5,14)(7,15,9,11)(17,20,22,18)(19,26,28,23)(21,24,30,29)(25,27,32,31)Polycyclic presentation:
2^9|a^2=gi,c^2=gi,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,h^b=hi,d^c=dh,e^c=eg,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72316709, #72316713, #72316716, … |
| maximal normal subgroup of | #72316709, #72316713, #72316716, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49943);