Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501132 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1337 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 1337 conjugacy classes of subgroups — too many to draw. Only its 140 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,9,15,10,12,7,16)(8,11,17,13,18,14,19,20);(1,2)(3,4)(6,11)(7,9)(8,18)(12,14)(13,15)(16,20);(2,4)(5,7)(6,12)(8,17)(9,10)(13,20)(18,19);(5,7)(6,13)(8,19)(9,10)(11,16)(12,20)(14,15)(17,18)Polycyclic presentation:
2^9|a^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,e^d=ei,f^d=fhi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72213335, #72219496, #72219528, … |
| maximal normal subgroup of | #72213335, #72219496, #72219528, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408328);