Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #108026 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22⋊#95 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4,7,8)(5,6)(9,10,14,19,15,20,13,18)(11,21,24,34,23,16,29,31)(12,25,28,17,26,33,27,32)(22,36,37,30,38,40,39,35);(1,3)(2,4)(5,7)(6,8)(9,11)(10,16,20,21)(12,22)(13,29)(14,24)(15,23)(17,30,32,35)(18,34,19,31)(25,36,33,40)(26,38)(27,39)(28,37);(1,4)(2,3)(5,8)(6,7)(9,12)(10,17)(11,22)(13,28)(14,27)(15,26)(16,30)(18,33)(19,25)(20,32)(21,35)(23,38)(24,39)(29,37)(31,36)(34,40)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=ehi,f^b=fh,d^c=dhi,e^c=ei,f^c=fhi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72235178, #72235200, #72235212, … |
| maximal normal subgroup of | #72235178, #72235200, #72235212, … |
| maximal quotient of | #72235868 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15232);