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 | C2×#2404 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3742 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)(5,6)(7,9)(8,11)(10,12)(13,14)(15,19)(16,17)(18,20)(21,22,25,24)(23,27,28,26);(6,7)(8,12)(14,17)(18,19)(21,23,25,28)(22,26,24,27);(8,12)(10,11)(14,17)(18,19)(23,28)(26,27);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,15,16,20)(14,18,17,19)(21,24,25,22)(23,26,28,27);(5,8)(6,10)(7,11)(9,12)(13,14)(15,19)(16,17)(18,20)(21,23)(22,26)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=f,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 | #72232344, #72232353, #72232411 |
| maximal normal subgroup of | #72232344, #72232353, #72232411 |
| maximal quotient of | #72323539, #72330750 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3768173);