Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417571 |
| 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 | C22×#453 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2294 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)(7,13,15,10)(8,16)(9,11)(12,17)(14,19,18,20);(7,14)(8,17)(10,19)(12,16)(13,20)(15,18);(5,7)(6,10)(8,18)(9,15)(11,13)(12,20)(14,17)(16,19);(2,4)(8,17)(12,16)(14,18)(19,20)Polycyclic presentation:
2^9|a^2=eh,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72219643, #72219654, #72219666, … |
| maximal normal subgroup of | #72219643, #72219654, #72219666, … |
| maximal quotient of | #72262000 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356783);