Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455822 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
901 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,11,15,24)(8,18,19,28)(9,14)(10,13)(12,20,27,22)(16,31)(17,26,30,34)(21,23)(25,33)(29,36)(32,35);(5,7,9,17,10,15,21,30)(6,11,23,34,13,24,14,26)(8,16,22,33,19,29,20,32)(12,25,18,31,27,35,28,36);(1,2,3,4)(7,15)(9,21)(11,24)(12,27)(14,23)(20,22)(25,35)(32,33);(5,8)(6,12)(7,16)(9,22)(10,19)(11,25)(13,27)(14,28)(15,29)(17,33)(18,23)(20,21)(24,35)(26,36)(30,32)(31,34)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,d^c=de,h^c=hi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381216);