Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2678 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3308 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)(7,11)(10,12)(13,14)(15,18)(16,17)(19,20)(21,22,25,28)(23,27,24,26);(1,2,4,3)(5,6,9,8)(7,10,11,12)(14,17)(18,20)(21,22,25,28)(23,26,24,27);(2,3)(13,15,16,19)(14,18,17,20)(21,23,25,24)(22,26,28,27);(5,7)(6,10)(8,12)(9,11)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,27)(25,28);(1,3,4,2)(5,8,9,6)(7,12,11,10)(13,15)(14,18)(16,19)(17,20)(21,24,25,23)(22,27,28,26)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=fg,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72331981, #72332008, #72332041, … |
| maximal normal subgroup of | #72331981, #72332008, #72332041, … |
| maximal quotient of | #72332965 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2411139);