Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501196 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24404 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1106 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)(3,8)(4,14,16,7)(5,9)(6,10)(11,24)(12,25)(13,23,15,20)(17,26,31,21)(18,28)(19,27)(22,29,32,30);(2,7)(3,11)(5,18)(8,22)(9,26)(10,14)(12,19)(13,29)(15,30)(17,31)(20,24)(21,28)(23,27)(25,32);(1,3,6,12)(2,8,10,25)(4,15,16,13)(5,19,18,11)(7,20,14,23)(9,27,28,24)(17,29,31,30)(21,32,26,22);(2,7)(3,12)(8,23)(9,21)(10,14)(11,19)(13,15)(20,25)(22,27)(24,32)(26,28)(29,30)Polycyclic presentation:
2^9|a^2=f,c^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,f^b=fh,g^b=gi,d^c=di,e^c=ehi,g^c=gi,f^d=fh,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72237519, #72237521, #72237544, … |
| maximal normal subgroup of | #72237519, #72237521, #72237544, … |
| maximal quotient of | #72311946, #72312115, #72312368, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408394);