Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | #44871 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(18,22)(20,27)(23,29)(24,28)(26,31)(30,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16);(9,11)(10,13)(12,16)(14,15)(17,18,21,24)(19,23,25,30)(20,28,27,22)(26,32,31,29);(3,8)(5,7)(11,16)(13,15)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,31)(28,32);(10,14)(13,15)(19,25)(23,30)(26,31)(29,32)Polycyclic presentation:
2^9|a^2=f,c^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
| maximal subgroup of | #72281211, #72282358, #72282847 |
| maximal normal subgroup of | #72281211, #72282358, #72282847 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6901659);