Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13343 |
| 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×D4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C22 | #1889 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C22×D4 ▷ 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)(5,6,10,15,9,14,23,31)(7,16,20,32,19,12,27,29)(8,13,22,30,21,11,24,26)(17,25,33,35,18,28,34,36);(2,4)(6,11)(7,17)(8,21)(13,14)(15,30)(18,19)(20,34)(22,24)(25,28)(26,31)(27,33)(35,36);(5,7)(6,12)(8,18)(9,19)(10,20)(11,25)(13,28)(14,16)(15,29)(17,21)(22,34)(23,27)(24,33)(26,35)(30,36)(31,32)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^2=i,g^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,f^c=fi,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 | #72250552, #72257225, #72257229, … |
| maximal normal subgroup of | #72250552, #72257225, #72257229, … |
| maximal quotient of | #72264908, #72264939 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54968);