Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| socle | C22 | C24×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6782 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=g,f^2=i,h^2=i,b^a=bg,c^a=ch,e^a=ei,h^a=hi,c^b=ci,d^c=dh,h^c=hi,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72269226, #72280165, #72280442, … |
| maximal normal subgroup of | #72269226, #72280165, #72280442, … |
| maximal quotient of | C2×#10576709 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483905);