Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #496254 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #21750 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #60825 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #3500 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
876 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)(3,12,16,23)(4,10,19,30)(5,9)(6,11)(7,15,26,20)(8,18,27,21)(13,29)(14,31)(17,24)(22,28)(25,32);(1,3)(2,7)(4,17)(5,20)(6,16)(8,24)(9,23)(10,29)(11,26)(12,28)(13,18)(14,19)(15,22)(21,32)(25,30)(27,31);(1,4)(3,13)(5,21)(6,19)(7,23)(8,27)(9,28)(12,26)(14,15)(16,32)(17,20)(18,22)(24,25)(29,31);(1,4,6,19)(2,8)(3,14,16,17)(5,18,22,21)(7,24)(9,10)(11,27)(12,25)(13,20,32,15)(23,29)(26,31)(28,30)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=dh,e^b=ei,f^b=fi,g^b=gi,d^c=di,e^c=ei,g^c=gh,f^d=fi,g^d=gi,g^f=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403450);