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) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(6,8)(9,10)(11,16)(12,14)(13,15)(17,18)(19,23)(20,22)(21,24);(1,2,4,3)(9,11,14,13)(10,15,12,16)(17,19,22,21)(18,23,20,24);(2,3)(11,13)(15,16)(17,18,22,20)(19,24,21,23);(5,6,7,8)(9,12,14,10)(11,15,13,16)(17,20,22,18)(19,23,21,24);(1,3,4,2)(9,13,14,11)(10,15,12,16)(17,21,22,19)(18,23,20,24)Polycyclic presentation:
2^9|b^2=h,c^2=fh,d^2=i,e^2=h,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 | #72261040, #72261173, #72261196, … |
| maximal normal subgroup of | #72261040, #72261173, #72261196, … |
| maximal quotient of | #72323524, #72323559, C2×#2319841, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2227037);