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 | C2×#2679 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3862 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)(5,6,9,12)(7,8,11,10)(13,14)(15,16)(17,19)(18,20)(23,27)(26,28);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,19)(17,20);(1,2)(3,4)(13,14)(15,16)(17,19)(18,20)(21,22,25,24)(23,26,27,28);(5,7)(6,8)(9,11)(10,12)(13,14)(15,18)(16,20)(17,19)(21,23)(22,26)(24,28)(25,27);(1,2)(3,4)(5,8)(6,11)(7,12)(9,10)(13,16,17,18)(14,15,19,20)(21,24,25,22)(23,28,27,26)Polycyclic presentation:
2^9|a^2=fgh,c^2=i,e^2=hi,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 | #72326650, #72326654, #72326656, … |
| maximal normal subgroup of | #72326650, #72326654, #72326656, … |
| maximal quotient of | #72333283 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1293166);