Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458126 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2537 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
944 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)(7,17)(8,22)(9,12)(10,15)(11,16)(13,20)(14,26)(18,34)(19,29)(21,27)(23,30)(24,31)(25,32)(28,35)(33,36);(6,12,16,15)(8,23)(13,17,29,27)(14,30,32,22)(18,35)(24,36)(25,26)(28,31,34,33);(5,7,9,20,11,21,10,19)(6,13,15,17,16,29,12,27)(8,24,25,18,23,36,26,35)(14,31,30,28,32,33,22,34);(2,4)(5,8,9,25,11,23,10,26)(6,14,15,30,16,32,12,22)(7,18,19,24,21,35,20,36)(13,28,27,31,29,34,17,33)Polycyclic presentation:
2^9|b^2=g,c^2=f,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 372250);