Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499873 |
| 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 | C22×Q8 | C24 |
| prime radical | C2×2−1+4 | C23 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
914 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 914 conjugacy classes of subgroups — too many to draw. Only its 142 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:
(1,2,5,8)(3,7,4,6)(10,15)(11,12)(13,21)(16,23)(17,19)(22,24);(1,3,5,4)(2,6,8,7)(10,16)(13,21)(15,23)(17,22)(18,20)(19,24);(9,10)(11,15)(12,19)(13,22)(14,17)(16,21)(18,24)(20,23);(1,4,5,3)(2,7,8,6)(9,11,14,12)(10,15,17,19)(13,18,21,20)(16,23,22,24)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,f^c=fi,g^c=gi,h^c=hi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | Q8×#2205 |
| maximal normal subgroup of | Q8×#2205 |
| maximal quotient of | #72268164, #72268166 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408171);