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 | |
3929 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)(9,10)(11,14)(12,13)(15,16)(17,18,22,23)(19,21,20,24);(1,2,3,4)(5,6,7,8)(10,13)(14,15)(18,23)(21,24);(2,4)(5,6)(7,8)(10,13)(14,15)(17,19,22,20)(18,24,23,21);(2,4)(9,11)(10,14)(12,16)(13,15)(17,20,22,19)(18,21,23,24);(1,2)(3,4)(9,10)(11,15)(12,13)(14,16)(17,21)(18,20)(19,23)(22,24)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=i,d^2=i,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 | #72282231, #72326292, #72326331, … |
| maximal normal subgroup of | #72282231, #72326292, #72326331, … |
| maximal quotient of | #72328599, #72332690 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3919977);