Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×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 | |
5026 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)(5,6,8,9)(7,11,12,10)(14,17)(15,19)(21,22,25,27)(23,24,28,26);(6,9)(10,11)(13,14)(15,20)(16,17)(18,19)(21,23,25,28)(22,26,27,24);(7,12)(10,11)(14,17)(18,20)(22,27)(23,28);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,14)(15,20)(16,17)(18,19)(21,23)(22,24)(25,28)(26,27);(5,6)(7,11)(8,9)(10,12)(13,15)(14,18)(16,19)(17,20)(21,24)(22,28)(23,27)(25,26)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72230894, #72232325, #72232335, … |
| maximal normal subgroup of | #72230894, #72232325, #72232335, … |
| maximal quotient of | #72278217, #72278555, #72278848, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2977354);