Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61109 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D42 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| socle | C2 | C22×D42 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
10203 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
2^9|f^2=i,g^2=i,h^2=i,b^a=bg,c^a=ch,e^a=ei,g^a=gi,h^a=hi,g^b=gi,d^c=dh,e^c=ei,h^c=hi,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72214082, #72214208, #72214632, … |
| maximal normal subgroup of | #72214082, #72214208, #72214632, … |
| maximal quotient of | C4×#61110, #72224796 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483373);