Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13005 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1874 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(6,12)(7,17)(8,20)(9,10)(13,28)(14,21)(15,26)(16,27)(18,33)(19,24)(22,23)(25,29)(30,36)(31,35);(5,6,10,15,11,16,9,14)(7,13,17,29,19,31,24,30)(8,21,20,27,23,26,22,12)(18,28,32,36,33,35,34,25);(1,3,4,2)(5,7)(6,13)(8,18)(9,24)(10,17)(11,19)(12,25)(14,30)(15,29)(16,31)(20,32)(21,28)(22,34)(23,33)(26,35)(27,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=hi,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,h^a=hi,d^b=dhi,h^b=hi,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72321483, #72321536, #72321585, … |
| maximal normal subgroup of | #72321483, #72321536, #72321585, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54424);