Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
5021 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)(5,6,8,10)(7,9,12,11)(13,14,16,18)(15,19,20,17)(22,26)(24,27);(5,6)(7,9)(8,10)(11,12)(13,14,16,18)(15,17,20,19)(21,22,25,26)(23,24,28,27);(13,14)(15,17)(16,18)(19,20)(21,23)(22,27)(24,26)(25,28);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,17)(16,18)(19,20)(21,23)(22,24)(25,28)(26,27);(5,7)(6,9)(8,12)(10,11)(13,15)(14,17)(16,20)(18,19)(21,24)(22,28)(23,26)(25,27)Polycyclic presentation:
2^9|a^2=i,b^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72260814, #72260826, #72260830, … |
| maximal normal subgroup of | #72260814, #72260826, #72260830, … |
| maximal quotient of | #72298169, #72328472, #72329901, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4774318);