Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #161742 |
| 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 | |
1150 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)(21,22,25,28,26,29,24,32)(23,27,31,36,30,34,33,35);(5,7,8,15,9,14,17,20)(6,10,12,13,11,18,16,19)(21,23,25,33,26,30,24,31)(22,27,32,36,29,34,28,35);(1,2,3,4)(6,11)(7,14)(12,16)(15,20)(23,30)(27,34)(31,33)(35,36);(21,24,26,25)(22,28,29,32)(23,31,30,33)(27,35,34,36)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=fhi,d^2=i,e^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ch,e^b=ei,g^b=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112049);