Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| 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 | #16249 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3310 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)(7,10)(8,11)(9,12)(14,18)(15,16)(22,26)(23,28);(5,7)(6,9)(8,12)(10,11)(13,14)(15,20)(16,19)(17,18)(21,22)(23,24)(25,26)(27,28);(1,2)(3,4)(13,15,17,16)(14,19,18,20)(21,23)(22,27)(24,26)(25,28);(5,7)(6,10)(8,12)(9,11)(13,16,17,15)(14,19,18,20)(21,24,25,27)(22,23,26,28);(1,2)(3,4)(5,6)(7,9)(8,11)(10,12)(13,14)(15,19)(16,20)(17,18)(21,23)(22,24)(25,28)(26,27)Polycyclic presentation:
2^9|c^2=i,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232901, #72232905, #72232979, … |
| maximal normal subgroup of | #72232901, #72232905, #72232979, … |
| maximal quotient of | #72328440, #72328507, #72330865 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1194381);