Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
3531 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(6,8)(10,13)(11,15)(17,18)(19,23)(20,24)(21,22);(1,2,3,4)(9,10,12,13)(11,16,15,14)(17,19)(18,20)(21,24)(22,23);(1,2,3,4)(11,15)(14,16)(18,22)(19,24);(1,2,3,4)(5,6)(7,8)(9,10)(11,16)(12,13)(14,15)(17,20)(18,19)(21,23)(22,24);(1,2)(3,4)(9,11,12,15)(10,14,13,16)(17,18)(19,23)(20,24)(21,22)Polycyclic presentation:
2^9|b^2=fh,c^2=gh,d^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72280903, #72281048, #72281112, … |
| maximal normal subgroup of | #72280903, #72281048, #72281112, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3776984);