Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #388456 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14520 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23400 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,13,6,23)(3,11,16,25,5,22,15,9)(4,17,21,10,20,28,19,29)(14,26,31,27,18,24,30,32);(1,3,6,15,8,5,7,16)(2,9,12,11,13,25,23,22)(4,14,19,30,20,18,21,31)(10,26,28,27,29,24,17,32);(1,4)(2,10,13,29)(3,14)(5,18)(6,21)(7,19)(8,20)(9,24,25,26)(11,27,22,32)(12,17,23,28)(15,31)(16,30);(1,5)(2,11)(3,8)(4,18)(6,15)(7,16)(9,12)(10,27)(13,22)(14,20)(17,24)(19,30)(21,31)(23,25)(26,28)(29,32)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=eh,e^2=i,f^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,g^b=gi,e^c=ei,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237047, #72237066, #72237071, … |
| maximal normal subgroup of | #72237047, #72237066, #72237071, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295654);