Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418488 |
| 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 | C22×#41 | 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)(5,6,11,15,10,16,24,14)(7,17,20,31,21,12,28,32)(8,22,13,9,25,27,33,26)(18,19,23,34,30,35,36,29);(5,7,10,21)(6,12)(8,23,25,36)(9,19)(11,28,24,20)(13,18,33,30)(14,32)(15,31)(16,17)(22,34)(26,35)(27,29);(5,8,24,33,10,25,11,13)(6,9,15,27,16,26,14,22)(7,18,28,36,21,30,20,23)(12,29,32,19,17,34,31,35);(2,4)(5,9)(6,13)(7,19)(8,15)(10,26)(11,22)(12,30)(14,25)(16,33)(17,18)(20,29)(21,35)(23,32)(24,27)(28,34)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=fi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348809);