Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342574 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×H(5,𝔽2) | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
18466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6,8,12,9,10,15,17)(7,11,14,18,13,16,19,20);(6,10)(8,15)(11,16)(14,19);(2,4);(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20);(7,13)(11,16)(14,19)(18,20)Polycyclic presentation:
2^9|a^2=h,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^b=hi,e^d=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72219299, #72219302, #72219303, … |
| maximal normal subgroup of | #72219299, #72219302, #72219303, … |
| maximal quotient of | #72233017, D4×#1783, #72238521, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277900);