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 | #23402 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1206 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)(3,13)(4,18,21,24)(5,9,23,20)(6,11)(7,12)(8,16)(10,25)(14,22,19,31)(15,29,30,28)(17,26)(27,32);(1,3,6,16,7,17,25,32)(2,8,10,13,12,27,11,26)(4,19,20,29,21,14,9,28)(5,22,18,30,23,31,24,15);(1,4,6,9,7,21,25,20)(2,5,10,24,12,23,11,18)(3,14,16,29,17,19,32,28)(8,22,13,15,27,31,26,30);(1,5,7,23)(2,9,12,20)(3,15,17,30)(4,10,21,11)(6,24,25,18)(8,14,27,19)(13,29,26,28)(16,31,32,22)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=eh,d^2=i,e^2=i,f^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 | #72251777, #72251783, #72251799, … |
| maximal normal subgroup of | #72251777, #72251783, #72251799, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295653);