Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129741 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | #2847 | C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
915 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 915 conjugacy classes of subgroups — too many to draw. Only its 96 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 — 3 generators on 24 points:
(2,6,7,8)(9,10,13,16,14,17,12,22)(11,18,21,23,20,15,19,24);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,18)(22,24);(1,3,5,4)(2,6,7,8)(9,12)(10,16)(11,19)(13,14)(15,23)(17,22)(18,24)(20,21)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=ei,f^b=fgh,d^c=di,f^c=fi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72266604, #72266607, #72266608 |
| maximal normal subgroup of | #72266604, #72266607, #72266608 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37095);