Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2057 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(18,23)(19,26)(20,21)(24,32)(25,29)(27,31);(9,10)(11,15)(12,14)(13,16)(17,18,20,29,22,25,21,23)(19,27,30,31,26,24,28,32);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,30)(21,28)(22,26)(23,31)(25,27)(29,32);(11,16)(13,15)(18,23)(19,26)(20,21)(24,32)(25,29)(27,31);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,20,22,21)(18,23,25,29)(19,28,26,30)(24,32,27,31)Polycyclic presentation:
2^9|b^2=fi,c^2=gh,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,e^d=ei,f^d=fi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72285313, #72313256, #72313593, … |
| maximal normal subgroup of | #72285313, #72313256, #72313593, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6997699);