Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24260 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,9,3)(4,16,21,17)(5,6,15,10)(7,12,8,13)(11,26,14,20)(18,29,30,28)(19,32,27,31)(22,24,23,25);(1,3,6,15,9,2,10,5)(4,17,19,32,21,16,27,31)(7,12,24,22,8,13,25,23)(11,28,18,20,14,29,30,26);(1,4)(2,11)(3,14)(5,18)(6,19)(7,20)(8,26)(9,21)(10,27)(12,16)(13,17)(15,30)(22,31)(23,32)(24,29)(25,28);(1,5,6,3,9,15,10,2)(4,18,19,14,21,30,27,11)(7,22,24,13,8,23,25,12)(16,20,31,29,17,26,32,28)Polycyclic presentation:
2^9|a^2=i,b^2=e,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,g^c=gi,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72236866, #72236879, #72236900, … |
| maximal normal subgroup of | #72236900, #72237521, #72237523, … |
| maximal quotient of | #72311876, #72311877, #72311887, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408009);