Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9914 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | #386 |
| prime radical | C2×#2397 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1688 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 1688 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,8,11,9,7,12,10)(13,14)(15,17)(16,19)(18,20);(5,7)(6,9)(8,10)(11,12)(14,17)(15,20)(18,19);(2,4)(15,20)(17,18)Polycyclic presentation:
2^9|a^2=g,d^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=dh,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72228199, #72228214 |
| maximal normal subgroup of | #72228199 |
| maximal quotient of | #72228196 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22355);