Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485663 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C24 |
| prime radical | #60048 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
941 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 941 conjugacy classes of subgroups — too many to draw. Only its 120 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 — 4 generators on 32 points:
(1,2,7,12)(3,15,18,27)(4,11,21,31)(5,9,17,10)(6,23,19,32)(8,14,25,13)(16,30,20,26)(22,29,24,28);(2,9)(3,16)(4,19)(5,17)(6,21)(10,12)(11,15)(13,28)(14,29)(18,20)(22,24)(23,30)(26,32)(27,31);(1,3,8,16)(2,10,14,28)(4,17,6,24)(5,19,22,21)(7,18,25,20)(9,13,29,12)(11,30,23,15)(26,32,27,31);(1,4,8,6)(2,11,14,23)(3,17,16,24)(5,20,22,18)(7,21,25,19)(9,26,29,27)(10,30,28,15)(12,31,13,32)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,c^b=ci,d^b=dgi,e^b=eg,f^b=fhi,e^c=eh,f^c=fi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72209813, #72209816, #72228060, … |
| maximal normal subgroup of | #72209813, #72228060, #72228064, … |
| maximal quotient of | #72227645 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392943);