Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420461 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2120 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)(5,6)(7,14)(8,19)(9,12)(10,13)(11,17)(15,18)(16,20);(5,7,8,17,10,18,9,16)(6,11,12,14,13,20,19,15);(5,6)(7,15,18,14)(8,19)(9,12)(10,13)(11,17,20,16);(2,4)(7,16)(8,9)(11,15)(12,19)(14,20)(17,18)Polycyclic presentation:
2^9|b^2=e,c^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,f^b=fi,e^c=ei,e^d=ei,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72239534, #72252460, #72252465, … |
| maximal normal subgroup of | #72239534, #72252460, #72252465, … |
| maximal quotient of | #72219669, C2×#422761, #72265574, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329957);