Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| 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 | |
1757 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,27)(20,26)(21,23)(22,25)(24,29)(28,31)(30,32);(1,3,5,2)(4,7,8,6)(9,11,12,16)(10,13,14,15)(18,23)(19,28)(20,22)(24,31)(27,32)(29,30);(10,14)(13,15)(17,19)(18,24)(20,28)(21,29)(22,30)(23,27)(25,31)(26,32);(2,3)(6,7)(11,16)(13,15)(17,20,21,22)(18,25,23,26)(19,28,29,30)(24,31,27,32);(1,4,5,8)(2,6,3,7)(17,20,21,22)(18,26,23,25)(19,28,29,30)(24,32,27,31)Polycyclic presentation:
2^9|b^2=fi,d^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72282054 |
| maximal normal subgroup of | #72282054 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7419755);