Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #163983 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×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 | #35292 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1301 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,14)(8,12)(9,18)(10,13)(11,15)(16,19)(17,20)(21,22,24,28,25,27,29,35)(23,26,31,34,30,33,32,36);(5,7,8,17,10,16,9,15)(6,11,18,19,13,20,12,14)(21,23,29,31,25,30,24,32)(22,26,28,36,27,33,35,34);(2,4)(5,6)(7,11)(8,18)(9,12)(10,13)(14,15)(16,20)(17,19);(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,20,14)(23,30)(26,33)(31,32)(34,36)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=hi,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=dhi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 103981);