Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418769 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2782 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
545 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 545 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(7,8)(9,10)(11,17,24,19)(12,26)(13,31,25,33)(14,22)(15,21)(16,20)(18,28)(23,34,39,40)(27,32,38,35)(29,36)(30,37);(1,2,5,6)(3,7,4,8)(9,11,16,13,14,24,15,25)(10,17,20,31,22,19,21,33)(12,27,28,39,30,38,29,23)(18,34,37,32,36,40,26,35);(1,3,5,4)(2,7,6,8)(9,12)(10,18)(11,23)(13,27)(14,30)(15,29)(16,28)(17,32)(19,35)(20,37)(21,26)(22,36)(24,39)(25,38)(31,40)(33,34);(1,4,5,3)(2,7,6,8)(9,13,16,24,14,25,15,11)(10,19,20,33,22,17,21,31)(12,27,28,39,30,38,29,23)(18,35,37,34,36,32,26,40)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,c^2=egh,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361133);