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 | #60827 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #3498 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
882 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)(3,12,15,10)(4,16)(5,9)(6,11)(7,14,25,21)(8,13,28,18)(17,29,32,27)(19,26)(20,31)(22,24)(23,30);(1,3,6,15)(2,7,11,25)(4,17,20,32)(5,21,23,14)(8,16,28,31)(9,10,30,12)(13,22,18,19)(24,29,26,27);(1,4)(2,8)(3,13)(5,22)(6,20)(7,24)(9,29)(10,16)(11,28)(12,31)(14,32)(15,18)(17,21)(19,23)(25,26)(27,30);(1,3,6,15)(2,7)(4,18,20,13)(5,14,23,21)(8,26)(9,12)(10,30)(11,25)(16,29)(17,19,32,22)(24,28)(27,31)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=hi,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, 403453);