Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499819 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23900 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1962 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 1962 conjugacy classes of subgroups — too many to draw. Only its 148 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)(3,11)(4,8)(5,18)(6,10)(7,13)(9,19)(12,27)(14,26)(15,25)(16,23)(17,32)(20,29)(21,28)(22,30)(24,31);(1,3)(2,7)(4,13)(5,12)(6,14)(8,11)(9,20)(10,22)(15,19)(16,30)(17,28)(18,21)(23,26)(24,25)(27,32)(29,31);(3,12)(4,16)(5,19)(7,20)(9,18)(11,25)(13,29)(14,15)(21,22)(24,32)(26,27)(28,30);(2,8)(7,11)(9,24)(10,23)(18,32)(20,25)(21,27)(22,26)Polycyclic presentation:
2^9|g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,h^a=hi,c^b=cg,f^b=fh,g^b=gi,e^c=eh,g^c=gi,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72231746, #72236904, #72236905, … |
| maximal normal subgroup of | #72231746, #72237521, #72237523, … |
| maximal quotient of | C2×#499819, #72280636, #72280908, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407015);