Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| 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 | C2×C4×D4 |
| prime radical | #23567 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1186 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)(5,6,9,12)(7,11,8,10)(13,14,18,21,17,16,20,25)(15,19,24,27,22,23,26,28);(6,10)(7,8)(11,12)(13,15)(14,19)(16,23)(17,22)(18,24)(20,26)(21,27)(25,28);(2,4)(14,16)(15,22)(21,25)(24,26);(5,7,9,8)(6,11,12,10)(13,16,18,25,17,14,20,21)(15,23,24,28,22,19,26,27)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,c^b=ci,d^b=dh,e^b=eh,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72225125, #72318326, #72318344, … |
| maximal normal subgroup of | #72225125, #72318326, #72318344, … |
| maximal quotient of | #72225064, #72285843, #72286602, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 116508);