Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| socle | C23 | C2×#99 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
782 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 782 conjugacy classes of subgroups — too many to draw. Only its 210 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,6,8,11,9,12,7,10)(13,14)(15,19)(16,20,24,18)(17,21)(22,26,28,25)(23,27);(1,2,3,4)(14,18,21,20)(19,25,27,26);(5,7,9,8)(6,10,12,11)(13,15,17,23)(14,19,21,27)(16,22,24,28)(18,25,20,26)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=hi,d^2=i,e^2=h,b^a=bd,d^a=dg,f^a=fi,g^a=gi,e^b=egi,e^d=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72266618 |
| maximal normal subgroup of | #72266618 |
| maximal quotient of | #72266603 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34148);