Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417544 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3027 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2145 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)(6,10,12,16)(11,17,18,20);(8,15)(10,16)(11,18)(13,19);(5,6)(7,11)(8,16)(9,12)(10,15)(13,20)(14,18)(17,19);(2,4)(5,7)(6,11)(8,13)(9,14)(10,17)(12,18)(15,19)(16,20)Polycyclic presentation:
2^9|a^2=eh,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72233138, #72256325, #72324322 |
| maximal normal subgroup of | #72233138, #72256325, #72324322 |
| maximal quotient of | #72233126, #72237000, #72309789 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356779);