Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(3,4)(9,10)(11,15)(12,16)(13,14);(1,2)(3,7)(4,8)(5,6)(18,22)(19,25)(20,27)(23,30)(24,29)(26,31);(1,3,5,4)(2,7,6,8)(10,14)(15,16)(17,18,20,22,21,24,27,29)(19,23,25,28,26,30,31,32);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(19,26)(23,30)(25,31)(28,32);(2,6)(7,8)(9,12,13,11)(10,16,14,15)(17,19,20,25,21,26,27,31)(18,23,22,28,24,30,29,32)Polycyclic presentation:
2^9|c^2=fh,d^2=g,e^2=fgh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72313243, #72313428, #72313671, … |
| maximal normal subgroup of | #72313243, #72313428, #72313671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6750977);