Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62817 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C2×#3543 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2,4,3)(5,6,8,10)(7,9,11,12)(13,14,18,17)(15,16,20,19);(7,11)(9,12)(14,17)(15,20);(5,6)(7,9)(8,10)(11,12)(13,15,18,20)(14,19,17,16);(1,2,4,3)(13,16)(14,15)(17,20)(18,19);(1,3,4,2)(5,7,8,11)(6,9,10,12)(13,17,18,14)(15,16,20,19);(5,6)(7,9)(8,10)(11,12)(13,16)(14,15)(17,20)(18,19)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=hi,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh152 conjugacy classes — view the full character table.
| maximal subgroup of | #72234906, #72234908, #72234929, … |
| maximal normal subgroup of | #72234906, #72234908, #72234929, … |
| maximal quotient of | #72262124 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533529);