Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232138 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1036 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 202 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)(3,5)(4,6)(7,8)(9,10,14,21,15,22,16,20)(11,19,13,24,12,18,23,17);(1,3,5,2)(4,7,8,6)(9,11,15,12)(10,17,22,24)(13,16,23,14)(18,21,19,20);(1,4,5,8)(2,6,3,7)(9,12,16,23,15,11,14,13)(10,18,20,17,22,19,21,24);(2,3)(6,7)(9,13,14,11,15,23,16,12)(10,19,20,24,22,18,21,17)Polycyclic presentation:
2^9|a^2=h,b^2=efg,c^2=ef,d^2=fg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^c=hi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72339677, #72339686, #72339767 |
| maximal normal subgroup of | #72339677, #72339686, #72339767 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295092);