Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #63091 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C24×D4 |
| commutator, Frattini, cosocle | C4 | C27 |
| socle | C2 | C25×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#585 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
56113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
2^9|h^2=i,b^a=bh,g^a=gi,h^a=hi,g^b=gi,h^b=hi,f^c=fi,e^d=ei164 conjugacy classes — view the full character table.
| maximal subgroup of | #72227318, #72286611, #72319182, … |
| maximal normal subgroup of | #72227318, #72286611, #72319182, … |
| maximal quotient of | D8×#585, #72343696 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10494197);