Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419786 |
| 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 | |
1625 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)(3,15,18,11)(4,10)(5,9,17,24)(6,12)(7,14)(8,13)(16,26,32,30)(19,25,31,23)(20,27)(21,28)(22,29);(1,3)(2,9)(4,16)(5,6)(7,17)(8,18)(10,25)(11,12)(13,24)(14,15)(19,20)(21,32)(22,31)(23,28)(26,29)(27,30);(1,4)(2,10)(3,16)(5,19)(6,20)(7,22)(8,21)(9,23)(11,26)(12,27)(13,28)(14,29)(15,30)(17,31)(18,32)(24,25);(1,5,7,18,8,17,6,3)(2,11,14,24,13,15,12,9)(4,19,22,32,21,31,20,16)(10,26,29,25,28,30,27,23)Polycyclic presentation:
2^9|a^2=f,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72240432, #72240441, #72240455, … |
| maximal normal subgroup of | #72240432, #72240441, #72240455, … |
| maximal quotient of | #72251167, #72251297, C2×#419787, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326983);