Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393934 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#3505 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(13,14)(15,16,19,20)(17,18);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,19)(18,20);(2,4)(6,9)(10,12)(13,16)(14,15)(17,20)(18,19);(5,7,8,11)(6,10,9,12)(13,16)(14,15)(17,20)(18,19)Polycyclic presentation:
2^9|a^2=efi,d^2=g,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^b=fi,h^b=hi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72261197, #72281551 |
| maximal normal subgroup of | #72261197, #72281551 |
| maximal quotient of | #72261262, #72261683, #72281584, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311443);