Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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 |
| prime radical | #15258 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3492 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)(6,9)(10,12)(13,14,17,15)(16,18,20,19)(21,22)(23,28)(24,25)(26,27);(14,15)(18,19)(22,27)(23,24);(1,2)(3,4)(16,20)(18,19)(21,23,26,24)(22,28,27,25);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,19,20,18)(21,24,26,23)(22,28,27,25);(5,7,8,11)(6,10,9,12)(13,16)(14,18)(15,19)(17,20)(21,25,26,28)(22,23,27,24)Polycyclic presentation:
2^9|a^2=fgh,c^2=h,d^2=f,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232361, #72232362, #72232364, … |
| maximal normal subgroup of | #72232361, #72232362, #72232364, … |
| maximal quotient of | #72323565, #72323686, #72330585 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4309991);