Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #386085 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14419 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60644 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1632 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)(3,9,13,7)(4,15)(5,19)(6,10)(8,17)(11,21)(12,25,28,22)(14,23,20,27)(16,24,31,29)(18,26)(30,32);(1,3)(2,7)(4,16)(5,20)(6,13)(8,24)(9,10)(11,27)(12,18)(14,21)(15,29)(17,31)(19,23)(22,26)(25,30)(28,32);(1,4,5,18,6,17,21,32)(2,8,11,26,10,15,19,30)(3,12,14,16,13,28,20,31)(7,22,23,24,9,25,27,29);(1,3)(2,9,10,7)(4,12)(5,14)(6,13)(8,25,15,22)(11,27,19,23)(16,18)(17,28)(20,21)(24,26,29,30)(31,32)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,f^c=fi,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72240441, #72240447, #72240448, … |
| maximal normal subgroup of | #72240441, #72240447, #72240448, … |
| maximal quotient of | #72251169, #72255807 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293281);