Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682420 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1905 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,29)(27,32)(28,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15);(11,16)(13,15)(17,19,21,29,22,27,20,28)(18,23,24,26,25,31,30,32);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(19,27)(23,31)(26,32)(28,29);(2,5)(6,8)(17,20,22,21)(18,24,25,30)(19,28,27,29)(23,26,31,32)Polycyclic presentation:
2^9|c^2=g,d^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72256874, #72256891, #72256971 |
| maximal normal subgroup of | #72256874, #72256891, #72256971 |
| maximal quotient of | #72307902 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6692146);