Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13357 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1700 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1700 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,14,19,16,20,11,12)(13,18,21,23,15,24,22,17);(2,4)(9,11,16,14)(10,17,20,23)(12,18,19,24)(13,22,15,21);(6,8)(9,12)(10,11)(13,23)(14,20)(15,17)(16,19)(18,21)(22,24)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^f=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72212048, #72212208, #72212212, … |
| maximal normal subgroup of | #72212048, #72212208, #72212212, … |
| maximal quotient of | #72227407, #72266781, C2×#147778 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54974);