Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7389 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1333 |
| commutator | C16 | C2×C16 |
| Frattini, cosocle | #1311 | C22 |
| socle | C2 | #7070 |
| prime radical | #508 | C8 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C4×C16 ◁ #1311 ◁ G | |
| LECS | G ▷ #1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
87 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,11,7,12,28,31,8,14,4,17,21,20,22,10,9,15,23,30,6,19,27,25,29,13,24,32,26,3,16,18);(1,3)(2,6)(4,18)(5,25)(7,15)(8,12)(9,20)(10,24)(11,16)(13,21)(14,26)(17,28)(19,29)(22,30)(23,31)(27,32)Polycyclic presentation:
2^9|a^2=d,c^2=ei,d^2=f,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=ceg,d^b=dg,e^b=egi,f^b=fi,g^b=gi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214446, C2×#93759, #72224647, … |
| maximal normal subgroup of | #72214446, C2×#93759, #72224647, … |
| maximal quotient of | C2×#93759, #72216527, #72230713, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 955);