Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #380263 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14262 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1985 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)(5,9)(6,10)(12,21,27,19)(13,25,29,20)(15,22)(16,23)(17,30)(18,24)(26,32,28,31);(1,3)(2,7)(4,12)(5,13)(6,14)(8,21)(9,20)(10,11)(15,26)(16,27)(17,28)(18,29)(19,23)(22,32)(24,25)(30,31);(1,4)(2,8)(3,12)(5,17,18,15)(6,16)(7,19)(9,22,24,30)(10,23)(11,21)(13,28,29,26)(14,27)(20,31,25,32);(1,5)(2,9)(3,13)(4,15)(6,18)(7,20)(8,22)(10,24)(11,25)(12,26)(14,29)(16,17)(19,31)(21,32)(23,30)(27,28)Polycyclic presentation:
2^9|a^2=h,c^2=f,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237749, #72237761, #72237767, … |
| maximal normal subgroup of | #72237749, #72237761, #72237767, … |
| maximal quotient of | #72265866 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287459);