Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13357 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C22 | #1889 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1669 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1669 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)(7,16,20,31)(8,13,21,11)(9,14)(10,15)(12,19,29,23)(17,33)(18,35)(22,30,24,27)(25,32)(26,36)(28,34);(2,4)(6,11)(7,17)(8,21)(13,15)(14,30)(18,20)(19,36)(22,24)(23,34)(26,28)(27,32)(33,35);(5,7)(6,12)(8,18)(9,23)(10,20)(11,26)(13,28)(14,31)(15,29)(16,32)(17,21)(19,25)(22,34)(24,36)(27,35)(30,33)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^f=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72250315, #72257229, #72257236, … |
| maximal normal subgroup of | #72250315, #72257229, #72257236, … |
| maximal quotient of | #72264908, #72308082 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54972);