Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456507 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16283 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23439 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1143 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)(3,13,14,24,17,22,16,8)(4,18,21,25,20,26,19,9)(15,29,31,27,30,23,28,32);(1,3,6,17)(2,8)(4,15,20,30)(5,16,7,14)(9,23)(10,13)(11,22)(12,24)(18,32)(19,28,21,31)(25,29)(26,27);(3,14)(4,19)(5,7)(8,22)(9,18)(10,11)(13,24)(15,30)(16,17)(20,21)(23,29)(25,26);(1,4)(2,9)(3,15)(5,21)(6,20)(7,19)(8,23)(10,26)(11,18)(12,25)(13,27)(14,28)(16,31)(17,30)(22,32)(24,29)Polycyclic presentation:
2^9|a^2=ef,b^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72338621, #72338626, #72338632 |
| maximal normal subgroup of | #72338621, #72338626, #72338632 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363703);