Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | #36624 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(6,10,11,12)(13,14,16,19,17,20,15,18);(5,6,7,10,9,11,8,12)(14,18)(15,16)(19,20);(2,4);(5,7,9,8)(6,10,11,12)(13,15,17,16)(14,18,20,19)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^b=di,e^b=eh,g^b=gh116 conjugacy classes — view the full character table.
| maximal subgroup of | #72343963, #72343966, #72343976 |
| maximal normal subgroup of | #72343963, #72343966, #72343976 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 100665);