Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499831 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #52004 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1378 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 1378 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(12,19)(18,24);(2,5)(6,8)(10,14)(11,17)(12,19)(15,22)(16,20)(21,23);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,20)(13,16)(14,19)(17,23)(18,22)(21,24);(2,5)(6,8)(9,11,13,17)(10,15,16,23)(12,18,19,24)(14,21,20,22)Polycyclic presentation:
2^9|c^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72213210, #72213695, #72213704, … |
| maximal normal subgroup of | #72213210, #72213695, #72213704, … |
| maximal quotient of | #72245391, #72250920, C2×#500179, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407375);