Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129414 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C24×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1748 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 1748 conjugacy classes of subgroups — too many to draw. Only its 158 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)(9,10,13,17,14,18,11,19)(12,16,20,23,22,24,21,15);(5,6)(7,8)(9,11,14,13)(10,15,18,23)(12,21,22,20)(16,17,24,19);(2,4)(6,8)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=di,g^b=ghi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72212048, #72212208, #72212212, … |
| maximal normal subgroup of | #72212048, #72212208, #72212212, … |
| maximal quotient of | #72227408, #72266775, C2×#147565 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54761);