Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394470 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#393 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,10)(8,15,17,11)(9,12)(13,20,16,18)(14,19);(5,7)(6,10)(8,16,17,13)(9,14)(11,18,15,20)(12,19);(5,8)(6,11)(7,13)(9,17)(10,18)(12,15)(14,16)(19,20);(2,4)Polycyclic presentation:
2^9|a^2=e,b^2=e,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225620, #72225630, #72225632, … |
| maximal normal subgroup of | #72225620, #72225630, #72225632, … |
| maximal quotient of | #72232921, #72233090, #72234595, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301666);