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 |
| prime radical | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3135 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)(15,19)(16,20)(17,18)(22,25)(23,28);(5,6,9,10)(7,12,11,8)(14,18)(16,19)(21,22,24,25)(23,26,28,27);(6,10)(8,12)(13,15,17,20)(14,19,18,16)(21,23,24,28)(22,26,25,27);(1,2,3,4)(5,7,9,11)(6,8,10,12)(13,16)(14,15)(17,19)(18,20)(21,22)(23,26)(24,25)(27,28);(5,8)(6,7)(9,12)(10,11)(13,15)(14,19)(16,18)(17,20)(21,23)(22,27)(24,28)(25,26)Polycyclic presentation:
2^9|b^2=gh,c^2=fh,d^2=fhi,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 | #72332189, #72332221, #72332265 |
| maximal normal subgroup of | #72332189, #72332221, #72332265 |
| maximal quotient of | #72332969 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4009166);