Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500839 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#2246 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,11)(8,9)(10,12)(14,21)(15,23)(16,26)(17,27)(18,24)(20,22)(25,28);(1,2)(3,4)(6,9)(10,11)(13,14,18,15,20,23,17,21,19,16,27,25,22,28,24,26);(2,4)(13,15,24,14,22,26,27,28,19,25,17,16,20,21,18,23);(5,7,8,12)(6,10,9,11)(13,16,27,15,20,28,24,21,19,14,18,25,22,23,17,26)Polycyclic presentation:
2^9|b^2=f,c^2=fhi,d^2=eh,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,h^a=hi,c^b=cg,d^b=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72318484, #72319486, #72319567 |
| maximal normal subgroup of | #72318484, #72319486, #72319567 |
| maximal quotient of | #72322603 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 411194);