Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #34736 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2245 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,5,6,7)(9,10)(11,15,16,13)(12,14);(1,2,3,7,4,6,8,5)(9,10)(11,13)(12,14)(15,16);(9,11)(10,13)(12,16)(14,15);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=h,b^a=be,c^a=cf,f^a=fi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72267299, #72267442, #72281184 |
| maximal normal subgroup of | #72267299, #72267442, #72281184 |
| maximal quotient of | #72267265 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 101858);