Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508070 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #23903 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1187 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 1187 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) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,18)(11,15,20,30)(12,22,24,34)(13,25,28,17)(16,27,32,23)(19,35,36,40)(21,33,26,31)(29,38,39,37);(2,5)(6,8)(9,11)(10,15)(12,19)(13,26)(14,20)(16,29)(17,33)(18,30)(21,28)(22,35)(23,37)(24,36)(25,31)(27,38)(32,39)(34,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,16)(11,19)(13,27)(14,24)(15,29)(17,34)(18,32)(20,36)(21,38)(22,25)(23,28)(26,37)(30,39)(31,40)(33,35);(3,7)(6,8)(11,20)(15,30)(19,36)(21,26)(29,39)(31,33)(35,40)(37,38)Polycyclic presentation:
2^9|a^2=i,f^2=i,b^a=be,c^a=cf,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,d^c=de,f^c=fi,h^c=hi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72311336, #72311346, #72311356, … |
| maximal normal subgroup of | #72311336, #72311346, #72311356, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415288);