Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14717 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2025 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)(4,15,17,8)(5,9)(6,10)(11,19)(12,26,16,20)(13,22)(14,21)(18,25,30,24)(23,28)(27,32,29,31);(1,3)(2,7)(4,16,17,12)(5,14)(6,13)(8,20,15,26)(9,21)(10,22)(11,23)(18,29,30,27)(19,28)(24,31,25,32);(1,4)(2,8)(3,12)(5,18)(6,17)(7,20)(9,24)(10,15)(11,25)(13,16)(14,27)(19,30)(21,31)(22,26)(23,32)(28,29);(2,9)(5,19)(7,21)(8,24)(10,11)(14,28)(15,25)(18,30)(20,31)(22,23)(26,32)(27,29)Polycyclic presentation:
2^9|a^2=e,b^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237642, #72240432, #72240443, … |
| maximal normal subgroup of | #72237642, #72240432, #72240443, … |
| maximal quotient of | #72237660, C2×#394484 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301680);