Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #388990 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14533 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | #23444 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1424 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)(3,11,14,7)(4,8,16,21)(5,9)(6,10)(12,18)(13,19,27,22)(15,20,29,31)(17,23)(24,26)(25,32)(28,30);(1,3)(2,7)(4,12)(5,13)(6,14)(8,18)(9,22)(10,11)(15,28)(16,26)(17,27)(19,23)(20,30)(21,24)(25,29)(31,32);(1,4)(2,8,10,21)(3,12)(5,15)(6,16)(7,18,11,24)(9,20,23,31)(13,25)(14,26)(17,29)(19,30,22,32)(27,28);(1,5)(2,9)(3,13)(4,15)(6,17)(7,19)(8,20)(10,23)(11,22)(12,25)(14,27)(16,29)(18,30)(21,31)(24,32)(26,28)Polycyclic presentation:
2^9|a^2=h,c^2=fh,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,g^d=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237746, #72237759, #72237768, … |
| maximal normal subgroup of | #72237746, #72237759, #72237768, … |
| maximal quotient of | #72265872 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296186);