Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #357 |
| commutator | C16 | C2×C16 |
| Frattini, cosocle | C8×C16 | C22 |
| socle | C2 | #7068 |
| prime radical | #508 | C8 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C16 ◁ C4×C16 ◁ C8×C16 ◁ G | |
| LECS | G ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
94 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,11,7,12,27,22,8,14,25,31,30,20,32,26,9,15,28,17,23,19,24,10,29,13,4,21,6,3,16,18);(1,3)(2,7)(4,22)(5,26)(6,13)(8,12)(9,20)(10,25)(11,24)(14,30)(15,23)(16,31)(17,27)(18,28)(19,29)(21,32)Polycyclic presentation:
2^9|a^2=d,c^2=egi,d^2=f,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=cei,d^b=di,e^b=eg,g^b=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72214147, #72214196, #72214214, … |
| maximal normal subgroup of | #72214147, #72214196, #72214214, … |
| maximal quotient of | C2×#93752, #72216535, #72226997 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 948);