Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #383184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14358 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3019 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1929 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)(4,18,21,9)(5,22)(6,11)(7,12)(8,16)(10,20,30,23)(14,26,15,29)(17,28)(19,25,32,31)(24,27);(1,3,5,16,7,17,6,24)(2,8,11,13,12,27,22,28)(4,19,20,14,21,32,23,15)(9,29,30,31,18,26,10,25);(1,4)(2,9)(3,14)(5,20)(6,23)(7,21)(8,25)(10,22)(11,30)(12,18)(13,29)(15,17)(16,32)(19,24)(26,28)(27,31);(1,4)(2,10)(3,15)(5,23)(6,20)(7,21)(8,26)(9,22)(11,18)(12,30)(13,31)(14,17)(16,32)(19,24)(25,28)(27,29)Polycyclic presentation:
2^9|a^2=h,b^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,e^d=ei,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72251777, #72251787, #72251798, … |
| maximal normal subgroup of | #72251777, #72251787, #72251798, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290380);