Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C43 |
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 | |
3526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,8)(9,10)(11,12)(14,15)(18,19)(21,22)(23,27)(24,25)(26,28);(6,10)(8,12)(13,14,17,15)(16,19,20,18)(21,23,25,28)(22,26,24,27);(1,2,4,3)(6,10)(7,11)(14,15)(18,19)(22,24)(26,27);(5,7,9,11)(6,8,10,12)(13,15,17,14)(16,19,20,18)(21,24,25,22)(23,27,28,26);(1,3,4,2)(5,8)(6,11)(7,10)(9,12)(13,16)(14,18)(15,19)(17,20)(21,23)(22,26)(24,27)(25,28)Polycyclic presentation:
2^9|b^2=i,c^2=fgh,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232465, #72232469, #72232552, … |
| maximal normal subgroup of | #72232465, #72232469, #72232552, … |
| maximal quotient of | #72331589, #72332982 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1032865);