Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11541 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C24 | C22×C8 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
914 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 914 conjugacy classes of subgroups — too many to draw. Only its 114 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:
(1,2)(3,4)(5,6,8,11,9,10,12,7)(13,14)(15,23)(16,20,19,25)(17,21,24,28)(18,22)(26,27);(5,7,8,10,9,11,12,6)(13,15,24,20,18,26,17,25)(14,19,28,23,22,16,21,27);(2,4)(13,16)(14,20)(15,21)(17,23)(18,19)(22,25)(24,27)(26,28)Polycyclic presentation:
2^9|a^2=f,b^2=df,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dg,e^b=ei,f^b=fg,g^b=gi,d^c=di,f^c=fi,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72236810 |
| maximal normal subgroup of | #72236810 |
| maximal quotient of | #72238405 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44888);