Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #160780 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,11,12,10)(8,9)(13,14)(15,17)(16,18)(19,20);(1,2,3,4)(6,9)(10,11)(14,17)(18,19);(5,7)(6,10)(8,12)(9,11)(14,18)(16,20)(17,19);(6,9)(10,11)Polycyclic presentation:
2^9|a^2=g,b^2=eh,f^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,f^c=fi,g^c=ghi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264124, #72264151, #72264156, … |
| maximal normal subgroup of | #72264124, #72264151, #72264156, … |
| maximal quotient of | C2×#188484 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95680);