Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #386038 |
| 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 | |
1607 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,13,6,11)(3,14,17,9,16,25,4,18)(5,20,21,10,22,27,23,28)(15,29,19,32,31,24,30,26);(1,3,8,16)(2,9,13,18)(4,6,17,7)(5,19,22,30)(10,26,28,32)(11,25,12,14)(15,23,31,21)(20,29,27,24);(1,4,7,3,8,17,6,16)(2,9,11,14,13,18,12,25)(5,19,23,15,22,30,21,31)(10,24,27,26,28,29,20,32);(1,5,6,21,8,22,7,23)(2,10,11,27,13,28,12,20)(3,15,4,19,16,31,17,30)(9,24,14,26,18,29,25,32)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=g,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,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 | #72213409, #72216059, #72216523, … |
| maximal normal subgroup of | #72213409, #72216059, #72216523, … |
| maximal quotient of | C2×#386041, #72250448, #72252202, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293237);