Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499724 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
991 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)(7,11)(8,9)(10,12)(14,18)(15,21)(16,23)(19,26)(20,27)(22,25)(24,28);(6,9)(10,11)(14,18)(15,21)(16,23)(19,26)(20,27)(22,25)(24,28);(2,4)(13,14)(15,22)(16,25)(17,19)(18,21)(20,23)(24,27)(26,28);(5,7,8,12)(6,10,9,11)(14,19)(18,26)(20,25)(22,27)Polycyclic presentation:
2^9|d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=di,f^c=fh,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225017, C2×#500616, #72319486, … |
| maximal normal subgroup of | #72225017, C2×#500616, #72319486, … |
| maximal quotient of | #72225000, #72225007, #72287692, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407812);