Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
3607 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,10)(7,11,12,8)(13,14)(15,16)(17,18)(19,20)(21,22,25,23)(24,26,28,27);(5,7,9,12)(6,8,10,11)(13,15)(14,19)(16,17)(18,20)(22,23)(26,27);(6,10)(8,11)(14,17)(15,20)(21,22,25,23)(24,27,28,26);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,16)(14,15)(17,20)(18,19)(21,23,25,22)(24,26,28,27);(5,8,9,11)(6,12,10,7)(13,17,18,14)(15,19,20,16)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=gh,b^2=fh,c^2=fg,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232597, #72232610, #72232655, … |
| maximal normal subgroup of | #72232597, #72232610, #72232655, … |
| maximal quotient of | #72328802, #72329044, #72330632 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3943845);