Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196092 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3038 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,11,14)(7,10,13,9)(8,15)(12,16)(17,19)(18,20);(5,7,9,6,11,13,10,14)(8,15,17,12,18,20,16,19);(1,2,3,4);(5,8)(6,12)(7,15)(9,17)(10,16)(11,18)(13,20)(14,19)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,c^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^b=gi,e^d=ei,g^d=gi,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224749, #72224754, #72224757, … |
| maximal normal subgroup of | #72224749, #72224754, #72224757, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300256);