Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #434722 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15506 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,21,7,10,6,11)(3,12,14,24,16,23,15,8)(4,17,19,25,20,31,18,9)(13,22,28,30,29,32,27,26);(1,3)(2,8)(4,13)(5,14)(6,15)(7,16)(9,22)(10,24)(11,23)(12,21)(17,30)(18,27)(19,28)(20,29)(25,32)(26,31);(1,4)(2,9,10,25)(3,13,16,29)(5,18)(6,19)(7,20)(8,22)(11,17,21,31)(12,26)(14,27,15,28)(23,30)(24,32);(2,10)(3,14)(5,6)(8,12)(9,25)(13,27)(15,16)(18,19)(22,26)(23,24)(28,29)(30,32)Polycyclic presentation:
2^9|a^2=fg,c^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237070, #72237073, #72237086, … |
| maximal normal subgroup of | #72237070, #72237073, #72237086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341918);