Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667980 |
| 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 | #30581 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1395 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)(3,6)(4,8)(5,7)(17,18,22,24,21,26,20,25)(19,23,29,32,27,31,28,30);(1,3,6,2)(4,8,5,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,28)(21,27)(22,29)(24,32)(25,30)(26,31);(9,11,12,16)(10,13,14,15)(18,24)(19,27)(20,22)(23,30)(25,26)(31,32);(1,4,6,5)(2,7,3,8)(19,27)(23,31)(28,29)(30,32);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(19,27)(23,31)(28,29)(30,32)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=hi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,g^c=gi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72313294, #72313344, #72313919 |
| maximal normal subgroup of | #72313294, #72313344, #72313919 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6668468);