Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4452694 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
4776 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)(5,6)(7,8)(9,10)(11,15)(12,13)(14,16)(18,22)(20,23);(5,6,7,8)(10,13)(14,15)(17,18)(19,20)(21,22)(23,24);(5,6,7,8)(9,10,12,13)(11,14,16,15)(17,19,21,24)(18,23,22,20);(1,2,3,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,24)(22,23);(9,11)(10,14)(12,16)(13,15)(17,20)(18,19)(21,23)(22,24)Polycyclic presentation:
2^9|b^2=fh,c^2=f,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72297229, #72297238, #72297282, … |
| maximal normal subgroup of | #72297229, #72297238, #72297282, … |
| maximal quotient of | #72298273, #72328482, #72329897, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4364664);