Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352469 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2983 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,9)(5,6)(10,27)(11,25)(12,26)(13,21)(14,17)(15,18)(20,22)(24,28)(29,30)(31,32);(1,2)(3,10)(4,11)(5,9)(6,21)(7,13)(8,12)(14,24)(15,25)(16,28)(17,29)(18,27)(19,30)(20,26)(22,32)(23,31);(1,3)(2,10)(4,7)(5,14)(6,17)(8,16)(9,24)(11,13)(12,28)(15,22)(18,20)(19,23)(21,29)(25,32)(26,27)(30,31);(1,4,7,3)(2,11,13,10)(5,15,6,18)(8,19,23,16)(9,25,21,27)(12,30,31,28)(14,20,17,22)(24,26,29,32);(1,5,7,6)(2,9,13,21)(3,14,4,17)(8,20,23,22)(10,24,11,29)(12,26,31,32)(15,16,18,19)(25,28,27,30)Polycyclic presentation:
2^9|d^2=h,e^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259675);