Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437415 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23428 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2152 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)(3,6)(4,5)(7,8)(18,23)(20,21)(24,30)(25,29)(27,28)(31,32);(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,23)(21,29)(22,25)(26,31)(27,30)(28,32);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,27)(21,28)(22,26)(23,30)(25,31)(29,32);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(19,26)(24,31)(27,28)(30,32);(17,20,22,21)(18,23,25,29)(19,27,26,28)(24,30,31,32)Polycyclic presentation:
2^9|c^2=h,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72281074, #72281149, #72281421, … |
| maximal normal subgroup of | #72281074, #72281149, #72281421, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6368674);