Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 | #37650 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2064 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(18,22)(19,25)(20,28)(23,27)(26,31)(29,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,18,21,22)(19,25,27,23)(20,24,28,30)(26,32,29,31);(10,14)(11,15)(17,19,22,25,21,27,18,23)(20,29,30,31,28,26,24,32);(2,6)(5,7)(17,20,21,28)(18,24,22,30)(19,26,27,29)(23,31,25,32);(3,8)(5,7)(11,15)(13,16)(19,27)(20,28)(23,25)(24,30)Polycyclic presentation:
2^9|b^2=i,c^2=ghi,d^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=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257462, #72257516, #72264334 |
| maximal normal subgroup of | #72257462, #72257516, #72264334 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6669102);