Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | #34963 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1501 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)(6,9,11,12)(13,14,16,20,17,21,24,18)(15,19,23,27,22,26,28,25);(5,6,10,9,8,11,7,12)(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);(15,22)(19,26)(23,28)(25,27)Polycyclic presentation:
2^9|a^2=g,b^2=egh,e^2=h,g^2=i,b^a=be,c^a=cf,e^b=eh,g^b=gh,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72319163, #72319529, #72319541, … |
| maximal normal subgroup of | #72319163, #72319529, #72319541, … |
| maximal quotient of | #72321243, #72321500 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 100693);