Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #185517 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3537 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2707 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9,11,6)(4,12,13,7)(5,8)(10,14)(15,16)(17,18,21,23,22,26,20,25)(19,24,29,30,27,31,28,32);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(18,23)(20,21)(24,30)(25,26)(28,29)(31,32);(1,4)(2,7)(3,10)(5,13)(6,14)(8,12)(9,15)(11,16)(17,19)(18,24)(20,28)(21,29)(22,27)(23,30)(25,32)(26,31);(3,11)(6,9)(10,16)(14,15)(19,27)(24,31)(28,29)(30,32)Polycyclic presentation:
2^9|a^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,g^b=ghi,d^c=di,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72314382 |
| maximal normal subgroup of | #72314382 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92885);