Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
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 | |
3442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,7)(8,12)(13,14)(15,17)(16,18)(19,20)(22,25)(26,28);(5,6,9,7)(8,11,12,10)(13,14,17,15)(16,18,20,19)(23,27)(26,28);(8,12)(10,11)(14,15)(16,20)(21,22,24,25)(23,26,27,28);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,17,14)(16,19,20,18)(21,22)(23,26)(24,25)(27,28);(5,8,9,12)(6,10,7,11)(13,16,17,20)(14,18,15,19)(21,23,24,27)(22,26,25,28)Polycyclic presentation:
2^9|b^2=f,c^2=gh,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72232685, #72232693, #72232705, … |
| maximal normal subgroup of | #72232685, #72232693, #72232705, … |
| maximal quotient of | #72332880 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5387893);