Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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 | C2×#3100 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,7,11,9,12,8,10)(14,18,21,24)(15,22,23,25);(1,2,3,4)(6,10)(7,8)(11,12)(13,14,19,18,17,21,16,24)(15,20,22,26,23,27,25,28);(6,11)(7,8)(10,12)(13,15)(14,20)(16,25)(17,23)(18,26)(19,22)(21,27)(24,28);(14,21)(18,24)(20,27)(26,28)Polycyclic presentation:
2^9|a^2=g,b^2=egh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^b=dhi,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72286123, #72318352, #72318361, … |
| maximal normal subgroup of | #72286123, #72318352, #72318361, … |
| maximal quotient of | #72286382, #72286630, #72287017, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 100813);