Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13350 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#3199 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1167 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1167 conjugacy classes of subgroups — too many to draw. Only its 138 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 — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,15,18,28)(8,20,22,12)(9,13)(10,14)(11,17)(16,32,19,31)(21,27)(23,35,36,29)(24,30)(25,33)(26,34);(5,7)(6,11)(8,21)(9,16)(10,18)(12,28)(13,25)(14,27)(15,20)(17,22)(19,24)(23,34)(26,30)(29,31)(32,35)(33,36);(2,4)(7,16)(9,24)(11,25)(12,20)(13,30)(15,31)(17,33)(18,19)(21,34)(23,36)(26,27)(28,32)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,d^c=dh,f^c=fi,e^d=ei,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72311865, #72311877, #72311880, … |
| maximal normal subgroup of | #72311865, #72311877, #72311880, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54995);