Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #503229 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #25771 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1161 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1161 conjugacy classes of subgroups — too many to draw. Only its 146 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:
(2,7)(4,8)(5,17)(9,19)(10,26)(11,21)(12,27)(13,22)(18,20)(24,29)(25,31)(28,32);(1,2,8,21,6,11,4,7)(3,12,22,32,15,28,13,27)(5,18,23,20,17,10,16,26)(9,24,31,30,25,29,19,14);(1,3)(2,9)(4,13)(5,14,17,30)(6,15)(7,19)(8,22)(10,12,18,28)(11,25)(16,29,23,24)(20,27,26,32)(21,31);(1,2,8,21,6,11,4,7)(3,9,22,31,15,25,13,19)(5,10,23,26,17,18,16,20)(12,24,32,30,28,29,27,14)Polycyclic presentation:
2^9|b^2=ei,c^2=f,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72236997, #72237526, #72237552, … |
| maximal normal subgroup of | #72236997, #72237526, #72237552, … |
| maximal quotient of | #72311961, #72312166, #72312686, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410428);