Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485611 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #47031 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1018 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 1018 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:
(1,2,6,12)(3,15,19,26)(4,10,21,30)(5,8,17,11)(7,14,25,13)(9,24,28,18)(16,29,23,27)(20,31,22,32);(2,8)(3,16)(4,20)(5,17)(9,13)(10,29)(11,12)(14,28)(15,32)(18,24)(19,23)(21,22)(26,31)(27,30);(1,3,7,16)(2,9,14,11)(4,18,22,17)(5,21,24,20)(6,19,25,23)(8,12,28,13)(10,15,32,29)(26,31,27,30);(1,4)(2,10)(3,17)(5,19)(6,21)(7,22)(8,26)(9,29)(11,15)(12,30)(13,31)(14,32)(16,18)(20,25)(23,24)(27,28)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,c^b=ci,d^b=dgi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72232556, #72234734, #72234735, … |
| maximal normal subgroup of | #72232556, #72234734, #72234735, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392908);