Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #383887 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14365 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1561 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)(3,7,12,23)(4,15,18,25)(5,8)(6,10)(9,17,26,20)(11,19)(13,28)(14,24,27,21)(16,31)(22,29)(30,32);(1,3,5,14,6,12,19,27)(2,7,11,21,10,23,8,24)(4,16,17,30,18,13,20,29)(9,22,25,28,26,32,15,31);(2,8)(3,12)(5,19)(7,21)(9,25)(10,11)(13,16)(15,26)(17,20)(22,31)(23,24)(28,32);(1,4)(2,9)(3,13)(5,20)(6,18)(7,22)(8,25)(10,26)(11,15)(12,16)(14,30)(17,19)(21,31)(23,32)(24,28)(27,29)Polycyclic presentation:
2^9|a^2=h,b^2=f,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,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 | #72318354, #72318925, #72318991, … |
| maximal normal subgroup of | #72318354, #72318925, #72318991, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291083);