Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #489347 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #18524 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #61111 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
896 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 896 conjugacy classes of subgroups — too many to draw. Only its 124 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,3)(4,14,17,30)(5,19,22,20)(6,13,23,10)(8,25,11,16)(9,28,12,24)(15,29,18,31)(21,26,32,27);(1,3,7,2)(4,11,17,8)(5,20,22,19)(6,10,23,13)(9,24,12,28)(14,21,30,32)(15,26,18,27)(16,31,25,29);(1,4)(2,8)(3,11)(5,21)(6,15)(7,17)(9,29)(10,26)(12,31)(13,27)(14,19)(16,24)(18,23)(20,30)(22,32)(25,28);(5,22)(9,12)(14,30)(16,25)(19,20)(21,32)(24,28)(29,31)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,h^a=hi,e^b=eg,f^b=fh,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72214826, #72215869, #72306631, … |
| maximal normal subgroup of | #72214826, #72215869, #72306631, … |
| maximal quotient of | #72226757, #72258294, #72258315 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396546);