Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682564 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36504 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,32)(29,31);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14);(2,6)(3,8)(10,14)(15,16)(17,19,21,28,22,27,20,29)(18,23,24,26,25,31,30,32);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(19,27)(23,31)(26,32)(28,29);(9,12,13,11)(10,15,14,16)(17,20,22,21)(18,24,25,30)(19,28,27,29)(23,32,31,26)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72251266, #72251279, #72256971 |
| maximal normal subgroup of | #72251266, #72251279, #72256971 |
| maximal quotient of | #72308947 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6791270);