Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 | |
4710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,8)(9,10)(11,12)(13,14,17,19)(15,16,20,18)(21,22,24,27)(23,26,28,25);(6,10)(8,12)(13,15,17,20)(14,18,19,16)(21,23)(22,25)(24,28)(26,27);(1,2)(3,4)(6,10)(7,11)(15,20)(16,18)(21,23,24,28)(22,26,27,25);(5,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,20)(21,23)(22,26)(24,28)(25,27);(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,19)(18,20)(21,22)(23,26)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232591, #72232615, #72232619, … |
| maximal normal subgroup of | #72232591, #72232615, #72232619, … |
| maximal quotient of | #72237746, #72285845, #72293959, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4242839);