Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485590 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #18152 |
| commutator, Frattini, cosocle | C2×M16 | C24 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
684 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)(3,14)(4,8)(5,12)(6,10)(7,13)(9,16)(11,18)(15,28)(17,25)(19,24)(20,27)(21,23)(22,32)(26,31)(29,30);(2,8)(4,18)(5,6)(9,22)(10,23)(11,12)(13,24)(14,28)(15,30)(16,20)(19,21)(25,26)(27,29)(31,32);(1,3)(2,9)(4,15)(5,20)(6,16)(7,17)(8,22)(10,25)(11,28)(12,14)(13,27)(18,30)(19,31)(21,32)(23,26)(24,29);(2,10,13,12)(4,19)(8,23,24,11)(9,25,27,14)(15,31)(18,21)(22,26,29,28)(30,32)Polycyclic presentation:
2^9|d^2=h,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,g^a=gi,h^a=hi,e^b=eg,f^b=fhi,g^b=gi,e^c=eh,f^c=fi,e^d=ei,f^e=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72231710, #72231715, #72250033, … |
| maximal normal subgroup of | #72231710, #72231715, #72250033, … |
| maximal quotient of | #72250331, #72250336, #72250581, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392786);