Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C23×C44 |
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 | |
4164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,9)(8,10)(11,12)(13,14,16,17)(15,18,20,19)(21,22,25,26)(23,24,27,28);(2,4)(6,7)(10,11)(13,14)(15,18)(16,17)(19,20)(22,26)(23,27);(1,2,3,4)(8,12)(10,11)(14,17)(18,19)(21,23)(22,24)(25,27)(26,28);(5,7,9,6)(8,10,12,11)(13,15,16,20)(14,18,17,19)(21,23)(22,24)(25,27)(26,28);(5,8)(6,10)(7,11)(9,12)(13,14)(15,19)(16,17)(18,20)(21,24)(22,27)(23,26)(25,28)Polycyclic presentation:
2^9|a^2=fi,c^2=fhi,d^2=gi,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 | #72234676, #72234678, #72303510, … |
| maximal normal subgroup of | #72234676, #72234678, #72303510, … |
| maximal quotient of | #72328884 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3455790);