Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396118 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2420 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
976 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)(7,15)(8,22)(9,13)(10,17)(11,16)(12,18)(14,26)(19,34)(20,30)(21,29)(23,28)(24,36)(25,32)(27,33)(31,35);(5,7,10,9,12,21,11,20)(6,13,16,15,18,30,17,29)(8,23,27,35,26,24,25,19)(14,31,33,28,22,34,32,36);(5,8,10,25,12,26,11,27)(6,14,16,32,18,22,17,33)(7,19,9,24,21,35,20,23)(13,28,15,31,30,36,29,34);(2,4)(5,9,10,21,12,20,11,7)(6,15,16,30,18,29,17,13)(8,24,25,35,26,23,27,19)(14,31,32,36,22,34,33,28)Polycyclic presentation:
2^9|b^2=ef,c^2=e,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=di,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 304308);