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) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3256 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)(7,9)(8,11)(10,12)(13,14)(15,19)(16,18)(17,20)(22,24)(26,27);(1,2,3,4)(6,7)(8,12)(13,14,16,18)(15,19,20,17)(21,22)(23,27)(24,25)(26,28);(1,2,3,4)(8,12)(10,11)(15,20)(17,19)(22,24)(26,27);(1,2,3,4)(5,7,9,6)(8,10,12,11)(13,14)(15,19)(16,18)(17,20)(21,23,25,28)(22,26,24,27);(1,2)(3,4)(5,8,9,12)(6,10,7,11)(13,15,16,20)(14,17,18,19)(21,24,25,22)(23,26,28,27)Polycyclic presentation:
2^9|b^2=fghi,c^2=g,d^2=hi,e^2=fg,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 | #72232331, #72232364, #72232376, … |
| maximal normal subgroup of | #72232331, #72232364, #72232376, … |
| maximal quotient of | #72323504, #72329895, #72330946 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3875355);