Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419798 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15273 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,8,13,7,22)(3,11,15,24,16,25,5,9)(4,17,20,29,21,28,19,10)(14,23,18,32,31,26,30,27);(1,3)(2,9)(4,14)(5,6)(7,15)(8,16)(10,26)(11,22)(12,25)(13,24)(17,32)(18,19)(20,30)(21,31)(23,29)(27,28);(1,4)(2,10)(3,14)(5,18)(6,19)(7,20)(8,21)(9,23)(11,27)(12,28)(13,29)(15,30)(16,31)(17,22)(24,26)(25,32);(1,5,7,16,8,15,6,3)(2,11,22,9,13,25,12,24)(4,18,20,31,21,30,19,14)(10,27,17,23,29,32,28,26)Polycyclic presentation:
2^9|a^2=f,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215958, #72216029, #72216059, … |
| maximal normal subgroup of | #72215958, #72216029, #72216059, … |
| maximal quotient of | C2×#419799, #72255818, #72314082 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326995);