Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #466 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9072 |
| prime radical | #3067 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,13,7,14,22,31,8,15,4,21,5,12,23,11)(3,16,19,9,18,26,32,25,20,30,17,28,24,10,29,27);(1,3)(2,9,12,27,15,28,14,25)(4,19)(5,24)(6,17)(7,18)(8,20)(10,11,30,21,26,31,16,13)(22,29)(23,32);(1,3)(2,10)(4,19)(5,24)(6,17)(7,18)(8,20)(9,11)(12,30)(13,25)(14,16)(15,26)(21,27)(22,29)(23,32)(28,31)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,e^b=ei,f^b=fi,d^c=di,e^c=ei,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72214259, #72214714, #72214756, … |
| maximal normal subgroup of | #72214259, #72214714, #72214756, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44824);