Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
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 | C2×#447 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1221 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,8,11,9,7,10,12)(13,14)(15,19)(16,18)(17,21)(20,24)(22,26)(23,25)(27,28);(14,18)(16,24)(19,25)(20,21)(23,28)(26,27);(2,4)(13,15)(14,19)(16,23)(17,22)(18,25)(20,27)(21,26)(24,28);(5,7,8,12,9,6,10,11)(15,22)(19,26)(23,28)(25,27)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,e^b=eh,d^c=dh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72320625, #72320638, #72320669 |
| maximal normal subgroup of | #72320625, #72320638, #72320669 |
| maximal quotient of | #72321116, #72321344 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 116388);