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) = C26×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 | |
3397 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)(6,10)(7,11)(13,14,17,19)(15,16,20,18)(21,22,25,26)(23,27,28,24);(5,6,9,10)(7,12,11,8)(13,15)(14,18)(16,19)(17,20)(22,26)(24,27);(6,10)(8,12)(14,19)(15,20)(21,23,25,28)(22,27,26,24);(1,2)(3,4)(5,7,9,11)(6,8,10,12)(13,16,17,18)(14,20,19,15)(21,24)(22,23)(25,27)(26,28);(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,19)(18,20)(21,23)(22,27)(24,26)(25,28)Polycyclic presentation:
2^9|a^2=fgh,b^2=gh,c^2=fh,d^2=h,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 | #72332195, #72332221, #72332263 |
| maximal normal subgroup of | #72332195, #72332221, #72332263 |
| maximal quotient of | #72332979 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3951992);