Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×D4×D8 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2455 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(15,21)(16,23)(18,24)(20,26)(22,27)(25,28);(6,9)(7,11)(13,14,17,19)(15,18,22,25)(16,20,23,26)(21,24,27,28);(13,15,16,21,17,22,23,27)(14,18,20,24,19,25,26,28);(5,7,8,11)(6,10,9,12);(1,3,4,2)(6,9)(10,12)(14,19)(15,22)(20,26)(21,27)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=g,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72311834, #72311874, #72311937, … |
| maximal normal subgroup of | #72311834, #72311874, #72311937, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6311203);