Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499750 |
| 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 ◁ #2930 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1293 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)(3,12)(4,7)(5,20)(6,10)(8,13)(9,16)(11,19)(14,28)(15,30)(17,22)(18,27)(21,26)(23,31)(24,29)(25,32);(2,7)(4,16)(5,19)(8,12)(9,11)(10,22)(13,18)(15,21)(17,31)(20,23)(24,28)(25,30)(26,27)(29,32);(4,17)(7,22)(9,23)(12,28)(13,29)(16,31)(18,32)(26,30);(1,3)(2,8)(4,18)(5,21)(6,14)(7,12)(9,26)(10,24)(11,27)(13,16)(15,19)(17,32)(20,25)(22,28)(23,30)(29,31)Polycyclic presentation:
2^9|e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,h^a=hi,c^b=cg,e^b=eh,h^b=hi,e^c=ei,e^d=eh,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72213945, #72231603, #72250039, … |
| maximal normal subgroup of | #72213945, #72231603, #72250039, … |
| maximal quotient of | #72224933, #72237260, #72250387, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406946);