Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417542 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#3026 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2309 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(10,16)(13,19);(5,6)(7,10)(8,11)(9,12)(13,20)(14,16)(15,18)(17,19);(5,7)(6,10)(8,13)(9,14)(11,17)(12,16)(15,19)(18,20);(2,4)(5,8)(6,11)(7,13)(9,15)(10,17)(12,18)(14,19)(16,20)Polycyclic presentation:
2^9|a^2=eh,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^b=fi,e^c=ei,h^c=hi,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72234617, #72234644, #72234649, … |
| maximal normal subgroup of | #72234617, #72234644, #72234649, … |
| maximal quotient of | #72234621, #72237000, #72240914, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356776);