Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21782 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,11,14,9,8,10,13)(7,12,17,19,16,15,18,20);(1,2,3,4)(5,7)(6,12)(8,15)(9,16)(10,17)(11,18)(13,19)(14,20);(5,8,9,6)(7,15)(10,13,11,14)(12,16)(17,19)(18,20);(10,11)(13,14)(17,18)(19,20)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fgi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=ce,g^b=gi,e^c=ei,f^c=fi,g^c=gi,f^d=fi,g^d=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72219640, #72219668, #72219673, … |
| maximal normal subgroup of | #72219640, #72219668, #72219673, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262679);