Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485328 |
| 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 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
930 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 930 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) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(9,10)(11,18)(12,15)(13,21)(14,16)(17,20)(19,24)(22,23);(1,2)(3,6)(4,8)(5,7)(10,15)(11,13)(12,20)(16,17)(18,23)(21,24);(9,11,14,13)(12,19,20,22)(15,17)(23,24);(1,3,5,4)(2,6,7,8)(10,16)(15,17)(18,21)(23,24)Polycyclic presentation:
2^9|c^2=hi,d^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,e^c=eh,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72282385, #72308464, #72308486, … |
| maximal normal subgroup of | #72282385, #72308464, #72308486, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392659);