Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #94978 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#95 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,9,11,10,7,8,12)(13,14)(15,21,23,27)(16,19)(17,20)(18,22,25,28)(24,26);(5,7,10,6)(8,11,9,12)(13,15)(14,18,20,25)(16,22)(17,23)(19,21,26,27)(24,28);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=dh,f^b=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72284774, #72284793, #72285233, … |
| maximal normal subgroup of | #72284774, #72284793, #72285233, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13462);