Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501114 |
| 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 | |
1097 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)(4,9,15,7)(5,10)(6,11)(12,24)(13,25)(14,23,16,20)(17,26,31,21)(18,28)(19,27)(22,29,32,30);(2,7)(3,12)(4,15)(5,18)(8,22)(9,11)(10,26)(13,19)(14,30)(16,29)(20,24)(21,28)(23,27)(25,32);(1,3)(2,8)(4,16)(5,19)(6,13)(7,20)(9,23)(10,27)(11,25)(12,18)(14,15)(17,29)(21,32)(22,26)(24,28)(30,31);(2,7)(3,13)(4,15)(8,23)(9,11)(10,21)(12,19)(17,31)(20,25)(22,27)(24,32)(26,28)Polycyclic presentation:
2^9|a^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,d^c=di,e^c=eh,g^c=gi,e^d=ei,f^d=fhi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72237521, #72237523, #72237551, … |
| maximal normal subgroup of | #72237521, #72237523, #72237551, … |
| maximal quotient of | #72312047, #72312053, #72312082, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408310);