Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #162342 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #35293 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,13)(8,16)(9,11)(10,15)(12,17)(14,19)(18,20)(21,22,24,28,25,29,32,26)(23,27,31,35,30,34,36,33);(5,7,8,15,9,14,17,20)(6,10,12,13,11,18,16,19)(22,26)(24,32)(27,33)(28,29)(31,36)(34,35);(1,2,3,4)(6,11)(10,18)(12,16)(13,19)(21,23)(22,27)(24,31)(25,30)(26,33)(28,35)(29,34)(32,36);(7,14)(10,18)(13,19)(15,20)(23,30)(27,34)(31,36)(33,35)Polycyclic presentation:
2^9|a^2=g,b^2=egi,c^2=fh,e^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=ei,g^b=gi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112126);