Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499752 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23848 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2932 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1288 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)(4,10)(5,14)(6,12)(7,13)(8,16)(9,17)(11,21)(18,30)(19,31)(20,26)(22,27)(23,29)(24,28)(25,32);(1,3)(2,8)(4,19)(5,17)(6,24)(7,18)(9,27)(10,15)(11,28)(12,25)(13,26)(14,31)(16,21)(20,22)(23,32)(29,30);(1,3)(2,9)(4,20)(5,24)(6,17)(7,18)(8,11)(10,25)(12,30)(13,28)(14,15)(16,23)(19,22)(21,32)(26,27)(29,31);(2,10)(4,21)(5,6)(8,15)(9,25)(11,14)(12,27)(13,29)(16,19)(17,24)(20,32)(22,23)(26,30)(28,31)Polycyclic presentation:
2^9|e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=eh,h^b=hi,e^c=ei,f^c=fi,e^d=eh,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72213937, #72231603, #72231620, … |
| maximal normal subgroup of | #72213937, #72231603, #72231620, … |
| maximal quotient of | #72231549, #72237259, C2×#499752, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406948);