Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
798 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,8,11,9,7,10,12)(13,14,17,20)(15,21,22,28)(16,24,25,19)(18,26,27,23);(13,15,16,23,17,22,25,26)(14,18,19,21,20,27,24,28);(2,4);(5,7,8,12,9,6,10,11)(15,22)(18,27)(21,28)(23,26)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=gh,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=dh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72285836, #72285857, #72285998, … |
| maximal normal subgroup of | #72285836, #72285857, #72285998, … |
| maximal quotient of | #72286527, #72286564 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 117925);