Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498904 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
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 | |
2309 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 2309 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(6,11)(7,9)(8,16)(12,19)(13,20)(14,17);(5,6)(7,13)(8,17)(9,12)(10,14)(11,16)(15,20)(18,19);(2,4)(6,12)(7,9)(11,19)(13,14)(15,18)(17,20);(5,7,10,9)(6,13,14,12)(8,15,16,18)(11,19,17,20)Polycyclic presentation:
2^9|d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,g^a=ghi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=di,e^c=eh,g^c=gi,g^e=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72214565, #72231756, #72233839, … |
| maximal normal subgroup of | #72214565, #72231756, #72233839, … |
| maximal quotient of | #72214479, #72231752, #72233828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406105);