Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
5061 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)(5,6,9,7)(8,10,12,11)(13,14,16,18)(15,19,20,17)(23,27)(25,28);(6,7)(10,11)(15,20)(17,19)(21,22,24,26)(23,25,27,28);(5,6,9,7)(8,11,12,10)(13,15)(14,17)(16,20)(18,19)(21,23)(22,25)(24,27)(26,28);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,15)(14,17)(16,20)(18,19)(21,22)(23,25)(24,26)(27,28);(5,8)(6,10)(7,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,23)(22,25)(24,27)(26,28)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=f,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72230859, #72232582, #72232620, … |
| maximal normal subgroup of | #72230859, #72232582, #72232620, … |
| maximal quotient of | #72278515, #72278919, #72293743, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3243063);