Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #39416 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1725 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18)(19,26)(20,22)(21,24)(23,28)(25,30)(27,31)(29,32);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(18,20)(19,27)(24,25)(28,32);(2,6)(3,7)(10,13)(14,16)(17,19,21,28,22,27,30,32)(18,23,25,26,20,29,24,31);(9,11,12,15)(10,14,13,16)(17,20,22,18)(19,23,27,29)(21,24,30,25)(26,32,31,28);(3,7)(5,8)(17,21,22,30)(18,24,20,25)(19,28,27,32)(23,31,29,26)Polycyclic presentation:
2^9|c^2=fgh,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72240756, #72257520, #72264334 |
| maximal normal subgroup of | #72240756, #72257520, #72264334 |
| maximal quotient of | #72240780 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6670761);