Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C24 | C2×C4⋊C4 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(6,8)(13,14,16,20,18,22,17,21)(15,23,24,27,25,28,26,19);(2,4)(9,10)(11,12)(13,15,17,24,18,25,16,26)(14,19,21,23,22,27,20,28);(5,6,7,8)(10,12)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=di,g^b=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72214941, #72215705, #72215773 |
| maximal normal subgroup of | #72214941, #72215705, #72215773 |
| maximal quotient of | #72214880 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31213);