Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2804 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(10,12)(11,15)(17,18,21,24)(19,22,23,20)(26,28)(30,32);(1,2,4,6)(3,7,8,5)(9,10)(11,16)(12,13)(14,15)(17,19,21,23)(18,20,24,22)(27,31)(30,32);(3,8)(5,7)(10,12)(14,16)(17,19)(18,22)(20,24)(21,23)(25,26,29,28)(27,30,31,32);(1,2)(3,7)(4,6)(5,8)(9,11,13,15)(10,14,12,16)(17,19)(18,22)(20,24)(21,23)(25,27,29,31)(26,30,28,32);(1,3,4,8)(2,5,6,7)(9,12,13,10)(11,14,15,16)(17,20,21,22)(18,23,24,19)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=g,d^2=gi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72300732, #72300749, #72300789 |
| maximal normal subgroup of | #72300732, #72300749, #72300789 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3886907);