Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129437 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C2×#519 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1 | |
912 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 912 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:
(5,6,9,15,10,14,26,32)(7,16,19,34,18,12,24,30)(8,20,22,33,23,13,25,31)(11,27,29,21,28,35,36,17);(1,2)(3,4)(5,7)(6,11)(8,21)(9,24)(10,18)(12,13)(14,28)(15,29)(16,20)(17,23)(19,26)(22,27)(25,35)(30,31)(32,36)(33,34);(2,4)(5,7)(6,12,14,16)(8,17)(9,19)(10,18)(11,20,28,13)(15,30,32,34)(21,23)(22,27)(24,26)(25,35)(29,33,36,31)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=ei,g^b=ghi,e^c=ei,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54807);