Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #160780 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
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 | C26 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
4202 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,11,12,9)(8,10)(14,17)(15,19)(18,20);(5,7,8,12)(6,9,10,11)(13,14,16,18)(15,20,19,17);(2,4);(6,10)(9,11)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,e^b=ei,g^b=ghi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72311018, #72311040 |
| maximal normal subgroup of | #72311018, #72311040 |
| maximal quotient of | #72311535 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92243);