Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10577652 |
| 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 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6378 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
2^9|b^2=gi,c^2=i,f^2=h,g^2=i,b^a=bg,c^a=ch,e^a=eh,g^a=gi,e^b=ei,f^b=fi,d^c=dh98 conjugacy classes — view the full character table.
| maximal subgroup of | #72216080, #72268456, #72280787, … |
| maximal normal subgroup of | #72216080, #72268456, #72280787, … |
| maximal quotient of | #72284901, #72285021 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10488167);