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 | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
951 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,8,18,23)(9,24)(10,16)(11,20)(12,15)(13,22,19,29)(14,27)(17,33,25,36)(21,34,35,30)(26,31)(28,32);(5,7,10,19,12,18,11,13)(6,8,20,22,15,23,16,29)(9,25,26,34,28,17,27,30)(14,21,32,33,31,35,24,36);(5,8)(6,13,15,19)(7,16,18,20)(9,21)(10,22)(11,29)(12,23)(14,30,31,34)(17,32,25,24)(26,33)(27,36)(28,35);(2,4)(5,9,12,28)(6,14,15,31)(7,17,18,25)(8,21,23,35)(10,27,11,26)(13,30,19,34)(16,32,20,24)(22,36,29,33)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,c^2=e,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,h^c=hi,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, 340212);