Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533105 |
| 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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1824 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)(17,18,21,26)(19,25,22,23)(20,28,30,32)(24,29,31,27);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,22)(18,23,26,25)(20,27,30,29)(24,28,31,32);(10,13)(14,16)(17,20,21,30)(18,24,26,31)(19,27,22,29)(23,28,25,32);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16);(11,15)(14,16)(17,19,21,22)(18,25,26,23)(20,29,30,27)(24,28,31,32)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^c=ei,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6440878);