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) = C28 |
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 | |
4773 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,8)(9,10,12,13)(11,15,16,14)(17,18,21,24)(19,23,20,22);(1,2,3,4)(10,13)(11,16)(19,20)(22,23);(2,4)(11,16)(14,15)(17,19,21,20)(18,22,24,23);(2,4)(5,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,20,21,19)(18,23,24,22);(1,2)(3,4)(9,11)(10,14)(12,16)(13,15)(17,18)(19,22)(20,23)(21,24)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=g,d^2=g,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 | #72263365, #72263377, #72263409, … |
| maximal normal subgroup of | #72263365, #72263377, #72263409, … |
| maximal quotient of | #72298112, #72298379, #72329956, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3768571);