Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #389036 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14533 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23402 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,8,4,6,12)(3,13,16,9,18,14,17,19)(5,10,23,27,24,20,22,28)(15,29,21,32,30,26,31,25);(1,3)(2,9)(4,19)(5,21)(6,17)(7,16)(8,18)(10,26)(11,14)(12,13)(15,22)(20,29)(23,30)(24,31)(25,27)(28,32);(1,4,8,2)(3,14)(5,20,24,10)(6,12,7,11)(9,16)(13,18)(15,26)(17,19)(21,32)(22,28,23,27)(25,31)(29,30);(1,5)(2,10)(3,15)(4,20)(6,23)(7,22)(8,24)(9,25)(11,28)(12,27)(13,29)(14,26)(16,31)(17,21)(18,30)(19,32)Polycyclic presentation:
2^9|a^2=h,c^2=fhi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237066, #72237070, #72237082, … |
| maximal normal subgroup of | #72237066, #72237070, #72237082, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296235);