Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511856 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32899 |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #518 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2193 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)(5,9)(6,10)(7,14)(8,17)(12,23)(13,27)(16,30)(18,22)(19,26)(20,32)(21,31)(24,25)(28,29);(1,3)(2,7)(4,13)(5,14)(6,12)(8,20)(9,11)(10,19)(15,29)(16,28)(17,25)(18,26)(21,24)(22,23)(27,30)(31,32);(3,12)(5,18)(7,19)(11,23)(13,25)(16,31)(20,29)(24,27);(1,4,6,17)(2,8)(3,13,12,25)(5,16,18,31)(7,20)(9,21)(10,15)(11,24)(14,28,26,32)(19,29)(22,30)(23,27)Polycyclic presentation:
2^9|d^2=h,b^a=be,c^a=cf,d^a=dg,h^a=hi,c^b=ci,f^b=fhi,e^c=eh,g^d=gi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72233206, #72234419, #72240647, … |
| maximal normal subgroup of | #72233206, #72234419, #72240647, … |
| maximal quotient of | #72232813, #72232947, #72233135, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419052);