Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13068 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1893 |
| 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 ◁ C4 ◁ C22×C8 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,12,6,13,22,30,7,14,23,29,5,11,21,10)(3,15,16,8,17,25,32,24,19,31,20,26,18,9,27,28);(1,3)(2,8,13,24,14,26,11,28)(4,16)(5,18)(6,17)(7,19)(9,10,15,12,25,30,31,29)(20,23)(21,27)(22,32);(1,3)(2,9)(4,20)(5,18)(6,17)(7,19)(8,10)(11,31)(12,24)(13,15)(14,25)(16,23)(21,32)(22,27)(26,30)(28,29)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,e^b=ei,d^c=di,e^c=ei,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72214259, #72214424, #72214466, … |
| maximal normal subgroup of | #72214259, #72214424, #72214466, … |
| maximal quotient of | C2×#137627 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44823);