Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508104 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #29636 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #23990 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11)(3,12,16,30)(4,9,18,27)(5,7,19,10)(8,23,25,15)(13,21,31,28)(14,24,20,29)(17,22,32,26);(2,7)(5,19)(8,21)(9,22)(10,11)(12,24)(14,20)(15,23)(17,32)(25,28)(26,27)(29,30);(1,3)(2,8)(4,15)(5,20)(6,16)(7,21)(9,12)(10,28)(11,25)(13,17)(14,19)(18,23)(22,29)(24,26)(27,30)(31,32);(1,4,6,18)(2,9,11,27)(3,13)(5,17,19,32)(7,22,10,26)(8,24)(12,21)(14,15)(16,31)(20,23)(25,29)(28,30)Polycyclic presentation:
2^9|a^2=i,d^2=hi,e^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gh,c^b=cg,e^b=ei,f^b=fh,d^c=de,e^c=ei,h^c=hi,f^d=fi,g^d=gi,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415303);