Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10583022 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C24×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7881 conjugacy classes of subgroups.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
2^9|g^2=i,b^a=bg,c^a=ch,d^a=di,e^a=ei,g^a=gi,g^b=gi,f^c=fh,e^d=eh95 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,ℤ/8)×H(5,𝔽2), #72267627, #72316188, … |
| maximal normal subgroup of | AGL(1,ℤ/8)×H(5,𝔽2), #72267627, #72316188, … |
| maximal quotient of | #72282673, #72282693, #72282736, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10490261);