Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394267 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
4010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16);(9,10)(11,15,16,13)(12,14);(1,2,3,4)(9,11)(10,13)(12,16)(14,15);(6,8)Polycyclic presentation:
2^9|b^2=e,c^2=f,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,g^b=gi,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#2021, #72235078, #72235081, … |
| maximal normal subgroup of | D4×#2021, #72235078, #72235081, … |
| maximal quotient of | #72260761, #72260795, #72284380 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305106);