Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18132 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3006 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,11,12,9)(8,10)(13,14,16,18,17,19,15,20);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(14,18)(15,16)(19,20);(5,7)(6,9)(8,12)(10,11)(14,18)(15,16)(19,20);(14,19)(18,20)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fhi,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fh,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72278155, #72278973 |
| maximal normal subgroup of | #72278155, #72278973 |
| maximal quotient of | #72278451, #72278461, #72278468, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102322);