Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417690 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2419 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,8,14,10,15,9,13)(7,12,16,11,17,19,18,20);(1,2,3,4)(5,7,8,16,10,17,9,18)(6,11,13,12,15,20,14,19);(5,6)(7,12)(8,13)(9,14)(10,15)(11,18)(16,20)(17,19);(2,4)(5,7)(6,12)(8,18)(9,16)(10,17)(11,13)(14,20)(15,19)Polycyclic presentation:
2^9|a^2=f,b^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72213250, #72213323, #72213442, … |
| maximal normal subgroup of | #72213250, #72213323, #72213442, … |
| maximal quotient of | #72226270, C2×#418906 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326102);