Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C22×#127 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1914 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)(7,8)(9,12)(10,11)(13,14,18,19,17,15,16,20);(5,7,8,6,10,12,9,11)(13,15,18,20,17,14,16,19);(1,2,3,4)(13,16,17,18)(14,19,15,20);(5,8,10,9)(6,7,11,12)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ci,d^b=dh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72211656, C2×#204054, #72239306, … |
| maximal normal subgroup of | #72211656, C2×#204054, #72239306, … |
| maximal quotient of | #72211620, C2×#204054, #72239335 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111250);