Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #248839 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
4187 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(10,11)(14,17)(15,19)(18,20);(1,2)(3,4)(5,6,9,8)(7,10,12,11)(15,19)(17,20);(2,4)(5,7)(6,10)(8,11)(9,12)(13,14)(15,20)(16,18)(17,19);(5,8,9,6)(7,11,12,10)(15,19)(17,20)Polycyclic presentation:
2^9|b^2=hi,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^b=fh,e^c=eh,f^c=fi,f^d=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231238, #72234219, #72261482, … |
| maximal normal subgroup of | #72234219, #72261482, #72261897, … |
| maximal quotient of | #72231235, #72234240, #72279344, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 156087);