Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419839 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,15,11,16,29,35)(7,17,21,12,22,32,20,31)(8,23,26,28,14,13,25,9)(18,19,24,27,30,33,34,36);(5,7)(6,12,16,31)(8,24)(9,27,28,36)(10,20)(11,22)(13,33,23,19)(14,34)(15,17,35,32)(18,26)(21,29)(25,30);(5,8,11,14)(6,13,16,23)(7,18,22,30)(9,15,28,35)(10,25,29,26)(12,27,31,36)(17,33,32,19)(20,24,21,34);(2,4)(5,9,10,23,11,28,29,13)(6,14,35,26,16,8,15,25)(7,19,20,36,22,33,21,27)(12,30,32,34,31,18,17,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343809);