Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| 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 | #23924 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1977 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)(9,10)(11,14)(12,16)(13,15)(18,22)(19,25)(20,27)(23,30)(24,28)(29,31);(9,11,13,12)(10,14,15,16)(17,18,21,24)(19,23,25,31)(20,28,27,22)(26,30,32,29);(2,5)(6,8)(10,15)(11,12)(17,19,21,25)(18,23,24,31)(20,26,27,32)(22,29,28,30);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,15,14);(11,12)(14,16)(19,25)(23,31)(26,32)(29,30)Polycyclic presentation:
2^9|b^2=hi,c^2=i,d^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=eh83 conjugacy classes — view the full character table.
| maximal subgroup of | #72308651, #72308792, #72308971, … |
| maximal normal subgroup of | #72308651, #72308792, #72308971, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6518926);