Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2669 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(5,6,7,8)(9,10,12,13)(11,15,16,14)(17,18)(19,24)(20,22)(21,23);(10,13)(14,15)(17,19)(18,20)(21,22)(23,24);(1,2,4,3)(9,11)(10,14)(12,16)(13,15)(17,20)(18,22)(19,23)(21,24);(5,6)(7,8)(9,11)(10,15)(12,16)(13,14)(17,20)(18,19)(21,24)(22,23);(1,3,4,2)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,22)(21,23)Polycyclic presentation:
2^9|a^2=fi,c^2=fgh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72323397, #72323413, #72323418, … |
| maximal normal subgroup of | #72323397, #72323413, #72323418, … |
| maximal quotient of | #72323635, #72330519, C2×#1433994, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1341190);