Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454673 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16255 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60369 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1808 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,6,8,19,10)(3,12,13,23,16,21,15,7)(4,9,20,26,18,24,17,25)(14,29,28,22,30,27,31,32);(1,3)(2,7)(4,14)(5,13)(6,16)(8,23)(9,22)(10,21)(11,12)(15,19)(17,31)(18,30)(20,28)(24,32)(25,29)(26,27);(2,8)(3,13)(4,17)(5,19)(7,21)(9,25)(12,23)(14,30)(15,16)(18,20)(22,32)(24,26);(1,4)(2,9)(3,14)(5,20)(6,18)(7,22)(8,24)(10,25)(11,26)(12,27)(13,28)(15,31)(16,30)(17,19)(21,29)(23,32)Polycyclic presentation:
2^9|a^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72215936, #72216047, #72216059, … |
| maximal normal subgroup of | #72215936, #72216047, #72216059, … |
| maximal quotient of | C2×#454673, #72252200, #72252214, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361869);