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×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
969 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)(5,6,10,16,12,17,11,29)(7,18,20,13,21,14,8,22)(9,24,26,33,27,32,28,15)(19,35,23,31,36,30,25,34);(5,7)(6,13,17,22)(8,10)(9,25,27,23)(11,20)(12,21)(14,16,18,29)(15,31)(19,26,36,28)(24,35)(30,32)(33,34);(5,8,11,21,12,20,10,7)(6,14,16,22,17,18,29,13)(9,23,26,36,27,25,28,19)(15,31,32,35,33,34,24,30);(2,4)(5,9)(6,15)(7,19)(8,23)(10,28)(11,26)(12,27)(13,30)(14,31)(16,32)(17,33)(18,34)(20,25)(21,36)(22,35)(24,29)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=f,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, 348785);