Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485649 |
| 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×C4⋊C4 | C24 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1080 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 1080 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:
(2,5)(3,7)(6,8)(9,10)(11,15)(12,17)(13,16)(14,19)(18,22)(20,24)(21,23);(1,2)(3,8)(4,6)(5,7)(9,11,14,12)(10,15,19,17)(13,21,22,20)(16,24,18,23);(10,16)(11,12)(13,22)(15,23)(17,24)(18,19);(2,6)(5,8)(9,12,14,11)(10,17,19,15)(13,21,22,20)(16,23,18,24)Polycyclic presentation:
2^9|b^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=dg,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72339614, #72340418 |
| maximal normal subgroup of | #72339614, #72340418 |
| maximal quotient of | #72339636 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392870);