Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499831 |
| 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×D4 | C24 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1974 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1974 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,16)(8,12)(9,13)(10,15)(11,18)(14,24)(17,33)(19,31)(20,28)(21,30)(22,26)(23,29)(25,36)(27,34)(32,35);(1,2)(3,4)(5,7)(6,11)(8,17)(9,22)(10,19)(12,27)(13,16)(14,31)(15,26)(18,24)(20,36)(21,35)(23,34)(25,30)(28,33)(29,32);(2,4)(5,8,10,21)(6,12,15,30)(7,17,19,35)(9,23,24,20)(11,25,26,27)(13,28,14,29)(16,32,31,33)(18,36,22,34);(6,13)(9,24)(11,16)(12,28)(14,15)(18,22)(20,23)(25,32)(26,31)(27,33)(29,30)(34,36)Polycyclic presentation:
2^9|c^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fh,e^c=eh,e^d=ei,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72281791, #72282033, #72303059, … |
| maximal normal subgroup of | #72281791, #72282033, #72303059, … |
| maximal quotient of | #72282158, #72313430, #72339624 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407029);