Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419895 |
| 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 | C24 | H(5,𝔽2) |
| 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) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,18,23,32)(8,24)(9,26,19,34)(10,15)(11,17)(12,21,33,22)(13,25)(14,29,36,28)(16,30)(20,31)(27,35);(5,7,10,21,11,23,30,22)(6,12,16,32,17,33,15,18)(8,19,25,29,20,9,27,28)(13,34,24,14,35,26,31,36);(5,8,11,20)(6,13,17,35)(7,19,23,9)(10,27,30,25)(12,26,33,34)(14,18,36,32)(15,24,16,31)(21,28,22,29);(2,4)(5,9,11,19)(6,14)(7,20,23,8)(10,29,30,28)(12,31)(13,18)(15,26)(16,34)(17,36)(21,25,22,27)(24,33)(32,35)Polycyclic presentation:
2^9|a^2=efg,b^2=f,c^2=i,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 344165);