Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1955 |
| prime radical | C2×#525 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1042 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,15,10,16,25,33,11,13,24,29,8,14,22,31)(7,17,20,34,18,27,35,32,19,30,23,36,21,12,26,28);(2,4)(5,7)(6,12)(8,21)(9,23)(10,18)(11,19)(13,27)(14,30)(15,32)(16,17)(20,24)(22,35)(25,26)(28,29)(31,34)(33,36);(6,13)(9,24)(14,16)(20,23)(22,25)(26,35)(28,32)(34,36)Polycyclic presentation:
2^9|a^2=g,d^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=ei,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72281379, #72281410, #72281418, … |
| maximal normal subgroup of | #72281379, #72281410, #72281418, … |
| maximal quotient of | #72293221, #72293284 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54743);