Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #162118 |
| 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 | C24 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1090 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)(8,17)(9,11)(10,16)(12,14)(15,19)(18,20)(21,22,25,30,26,29,24,28)(23,27,33,35,32,36,31,34);(5,7,14,20,9,15,8,16)(6,10,17,19,11,18,12,13)(21,23,24,33,26,32,25,31)(22,27,30,34,29,36,28,35);(1,2,3,4)(6,11)(10,18)(12,17)(13,19);(7,15)(10,18)(13,19)(16,20)(21,24,26,25)(22,28,29,30)(23,31,32,33)(27,34,36,35)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=fhi,d^2=i,e^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=ei,g^b=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111998);