Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427820 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15375 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23445 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1304 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)(4,14)(5,18)(6,9)(8,17)(10,20)(11,29)(12,22)(13,27)(15,32)(16,28)(19,24)(21,26)(23,30)(25,31);(1,3)(2,7)(4,15)(5,19)(6,12)(8,25)(9,22)(10,27)(11,16)(13,20)(14,29)(17,30)(18,24)(21,31)(23,26)(28,32);(5,20)(7,22)(10,18)(13,19)(14,28)(17,21)(23,25)(30,31);(1,4)(2,8)(3,11,12,15)(5,21)(6,16)(7,23,22,25)(9,26)(10,28)(13,31,19,30)(14,18)(17,20)(24,32,27,29)Polycyclic presentation:
2^9|d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^c=hi,e^d=ei,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72285951, #72286231, #72286246, … |
| maximal normal subgroup of | #72285951, #72286231, #72286246, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335016);