Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #108534 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C4⋊C4 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1767 |
| prime radical | C23 | #423 |
| LCS | G ▷ C42 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
274 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 138 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:
(1,2,3,6,4,7,8,5)(10,15,17,23)(11,19)(13,22)(16,24,18,20);(2,5)(3,8)(6,7)(9,10,13,18,12,15,19,24,14,17,22,16,21,23,11,20);(9,11,12,13,14,19,21,22)(10,16,23,24,17,18,15,20)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=deg,d^2=gi,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ci,f^a=fi,c^b=ch,d^b=dg,e^b=egi,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72309415, D8×#347, #72309432, … |
| maximal normal subgroup of | #72309415, D8×#347, #72309432, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36305);