Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #23848 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1905 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)(10,14)(11,12)(17,18)(19,26)(20,22)(21,25)(23,29)(24,31)(27,30)(28,32);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(17,19,21,29)(18,23,25,26)(20,27,31,32)(22,28,24,30);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,20)(18,24)(19,27)(21,31)(22,25)(23,30)(26,28)(29,32);(1,4,5,3)(2,6,8,7)(9,11,13,12)(10,16,14,15)(18,25)(19,28)(22,24)(23,32)(26,27)(29,30);(2,8)(6,7)(9,12,13,11)(10,15,14,16)(18,25)(20,31)(23,26)(27,32)Polycyclic presentation:
2^9|b^2=f,c^2=fi,d^2=fi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72251178, #72251219, #72251252, … |
| maximal normal subgroup of | #72251178, #72251219, #72251252, … |
| maximal quotient of | #72264827, #72308824 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7483799);