Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2044 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #60753 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1115 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)(4,7)(5,11)(6,9)(8,15)(10,16)(13,26)(14,29)(17,21)(18,22)(19,24)(20,30)(23,32)(25,31)(27,28);(2,7)(3,13)(4,16)(8,20)(9,10)(11,21)(12,27)(17,31)(22,23)(24,25)(26,29)(30,32);(1,3,4,14,6,13,16,28)(2,8,10,23,9,22,7,20)(5,18,17,32,19,15,31,30)(11,26,25,29,24,12,21,27);(2,9)(7,10)(8,22)(11,24)(12,26)(20,23)(21,25)(27,29)Polycyclic presentation:
2^9|c^2=e,e^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,g^a=ghi,c^b=cg,e^b=ei,f^b=fh,f^c=fi,g^c=gi,h^c=hi,g^f=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406316);