Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500834 |
| 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 | #25779 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
851 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(14,21)(15,24)(16,27)(17,23)(18,25)(20,22)(26,28);(1,2)(3,4)(6,9)(10,12)(13,14,18,15,20,24,17,21,19,16,23,26,22,28,25,27);(2,4)(5,7,8,11)(6,10,9,12)(13,15,25,14,22,27,23,28,19,26,17,16,20,21,18,24);(6,9)(10,12)(13,16,18,26,20,28,17,27,19,14,23,15,22,24,25,21)Polycyclic presentation:
2^9|b^2=f,c^2=ei,d^2=f,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^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72319014, #72319524, #72319640, … |
| maximal normal subgroup of | #72319014, #72319524, #72319640, … |
| maximal quotient of | #72322637 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 412400);