Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511771 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1866 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 1866 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 12 points:
(2,4)(5,6)(7,9)(8,10)(11,12);(1,2,3,4);(6,9)(10,12);(7,11)Polycyclic presentation:
2^9|b^2=e,b^a=be,c^a=cf,d^a=dg,h^a=hi,g^c=ghi,f^d=fh,g^f=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72231273, #72231289, D4×#2196, … |
| maximal normal subgroup of | #72231273, #72231289, D4×#2196, … |
| maximal quotient of | #72234151, #72279452, C2×#511908, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419104);