Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #494749 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1898 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #60369 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#585 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1985 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)(3,6,14,23)(4,15,18,26)(5,19)(7,10)(8,11)(9,16,28,25)(12,29)(13,22,24,20)(17,31)(21,30)(27,32);(1,3,6,11,8,14,23,2)(4,12,21,28,18,27,17,9)(5,13,20,19,7,24,22,10)(15,25,31,32,26,16,30,29);(1,4)(2,9)(3,12)(5,16)(6,21)(7,25)(8,18)(10,26)(11,28)(13,30)(14,27)(15,19)(17,23)(20,29)(22,32)(24,31);(1,5,8,7)(2,10,11,19)(3,13,14,24)(4,16,18,25)(6,20,23,22)(9,26,28,15)(12,30,27,31)(17,32,21,29)Polycyclic presentation:
2^9|a^2=g,b^2=egi,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=eh,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72236949, #72237542, #72237549, … |
| maximal normal subgroup of | #72236949, #72237542, #72237549, … |
| maximal quotient of | #72312563, C2×#494752, #72313364, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401948);