Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573715 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
5261 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,9,12)(7,10,14,17)(8,11,15,18)(13,19,20,16);(1,2)(3,4)(5,7)(6,10)(8,13)(9,14)(11,16)(12,17)(15,20)(18,19);(2,4)(5,8)(6,11)(7,13)(9,15)(10,16)(12,18)(14,20)(17,19);(1,2)(3,4)(6,12)(10,17)(11,18)(16,19);(2,4)(6,12)(10,17)(11,18)(16,19)Polycyclic presentation:
2^9|a^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72240656, #72251173, #72251181, … |
| maximal normal subgroup of | #72240656, #72251173, #72251181, … |
| maximal quotient of | #72240670, #72267151, #72306553 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480922);