Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480674 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C24 |
| prime radical | C2×#2541 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1772 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 1772 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) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,4,8,5,6,3,7)(9,10)(11,14,16,13)(12,15);(2,6)(3,4)(10,13)(11,16)(14,15);(1,3,5,4)(2,7,6,8);(2,6)(7,8)Polycyclic presentation:
2^9|a^2=f,c^2=h,e^2=i,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=ch,e^b=ei,f^b=fgi,f^e=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72234181, #72234182, #72261951, … |
| maximal normal subgroup of | #72234181, #72234182, #72261951, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387923);