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×#3021 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,6,12,5,11)(3,13,17,23,16,28,15,8)(4,9,19,26,20,27,21,25)(14,29,18,22,30,24,31,32);(1,3,5,15,6,16,7,17)(2,8,10,13,12,23,11,28)(4,18,19,30,20,31,21,14)(9,24,25,22,27,29,26,32);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,22)(10,25)(11,26)(12,27)(13,29)(15,18)(16,30)(17,31)(23,32)(24,28);(1,3)(2,8)(4,14)(5,17)(6,16)(7,15)(9,22)(10,28)(11,13)(12,23)(18,21)(19,31)(20,30)(24,25)(26,29)(27,32)Polycyclic presentation:
2^9|a^2=h,b^2=e,e^2=i,g^2=i,h^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 | #72213603, #72213678, #72213687, … |
| maximal normal subgroup of | #72213603, #72213678, #72213687, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290382);