Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384297 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14376 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1120 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)(3,7)(4,8,16,23)(5,17,18,9)(6,10)(11,19,14,27)(12,25,26,20)(13,21)(15,29)(22,30)(24,32)(28,31);(1,3)(2,7)(4,14,16,11)(5,12)(6,13)(8,19,23,27)(9,20)(10,21)(15,24,30,28)(17,25)(18,26)(22,32,29,31);(1,4)(2,8,10,23)(3,11)(5,15)(6,16)(7,19,21,27)(9,22,17,29)(12,24)(13,14)(18,30)(20,31,25,32)(26,28);(1,5)(2,9)(3,12)(4,15)(6,18)(7,20)(8,22)(10,17)(11,24)(13,26)(14,28)(16,30)(19,31)(21,25)(23,29)(27,32)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=ef,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,h^c=hi,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237792, #72237795, #72237797, … |
| maximal normal subgroup of | #72237792, #72237795, #72237797, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291493);