Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3006 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,4)(5,6)(7,8)(9,10,12,14,13,11,16,15);(1,2,3,4)(9,11)(10,13)(12,14)(15,16);(9,10)(11,13)(12,14)(15,16);(6,8)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,e^b=ei,h^b=hi,f^c=fi98 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#32888, #72258359, #72264242, … |
| maximal normal subgroup of | C4×#32888, #72264242, #72264529, … |
| maximal quotient of | #72258359 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299840);