Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418535 |
| 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 | |
929 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)(5,6,11,16,12,9,10,15)(7,17,19,28,21,32,20,13)(8,22,25,24,27,31,26,14)(18,33,23,36,35,29,34,30);(5,7)(6,13,9,28)(8,23)(10,19)(11,20)(12,21)(14,30,24,36)(15,17,16,32)(18,25)(22,29,31,33)(26,35)(27,34);(5,8,11,25,12,27,10,26)(6,14,15,31,9,24,16,22)(7,18,19,23,21,35,20,34)(13,29,32,36,28,33,17,30);(2,4)(5,9)(6,12)(7,17)(8,24)(10,15)(11,16)(13,19)(14,27)(18,30)(20,28)(21,32)(22,25)(23,29)(26,31)(33,34)(35,36)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=fi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,e^d=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, 348786);