Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #380127 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14259 |
| 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 | |
1300 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)(3,11)(4,15,17,8)(5,9)(6,10)(7,14)(12,19,26,23)(13,24)(16,29,30,21)(18,22)(20,28)(25,31,27,32);(1,3)(2,7,10,11)(4,12)(5,13)(6,14)(8,19,15,23)(9,20,22,24)(16,25)(17,26)(18,28)(21,31,29,32)(27,30);(1,4)(2,8)(3,12)(5,16)(6,17)(7,19)(9,21)(10,15)(11,23)(13,27)(14,26)(18,30)(20,32)(22,29)(24,31)(25,28);(1,5)(2,9)(3,13)(4,16)(6,18)(7,20)(8,21)(10,22)(11,24)(12,25)(14,28)(15,29)(17,30)(19,31)(23,32)(26,27)Polycyclic presentation:
2^9|a^2=h,b^2=f,b^a=be,c^a=cf,f^a=fi,c^b=cg,e^b=ei,d^c=de,h^c=hi,g^d=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237786, #72237790, #72237801, … |
| maximal normal subgroup of | #72237786, #72237790, #72237801, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287323);