Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420406 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #35348 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2153 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(7,8)(9,10)(11,19,20,15)(12,16,21,24)(13,22)(14,18)(17,23);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,13)(14,20)(16,17)(18,19)(21,23)(22,24);(1,3,5,4)(2,7,6,8)(9,12)(10,16,18,24)(11,13)(14,21)(15,22,19,17)(20,23);(1,4,5,3)(2,7,6,8)(9,13)(10,17,18,22)(11,12)(14,23)(15,24,19,16)(20,21)Polycyclic presentation:
2^9|a^2=fg,b^2=eg,c^2=e,d^2=e,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72225699, #72225704, #72225708, … |
| maximal normal subgroup of | #72225699, #72225704, #72225708, … |
| maximal quotient of | #72284141 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348307);