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) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15258 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,6,7,8)(9,10,14,13)(11,16,12,15)(17,18,21,20)(19,22,23,24)(26,27)(28,32);(2,4)(6,8)(9,11,14,12)(10,15,13,16)(17,18)(19,22)(20,21)(23,24)(26,27)(30,31);(2,4)(6,8)(10,13)(15,16)(18,20)(22,24)(25,26,29,27)(28,30,32,31);(1,3,5,7)(2,6,4,8)(9,12,14,11)(10,16,13,15)(17,19)(18,22)(20,24)(21,23)(25,27,29,26)(28,31,32,30);(1,4,5,2)(3,6,7,8)(9,13,14,10)(11,15,12,16)(17,20,21,18)(19,22,23,24)(25,28,29,32)(26,30,27,31)Polycyclic presentation:
2^9|a^2=f,b^2=fgi,c^2=fg,d^2=fh,e^2=g,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 | #72326810, #72326857, #72326882, … |
| maximal normal subgroup of | #72326810, #72326857, #72326882, … |
| maximal quotient of | #72328866, #72330519, #72331073 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3790648);