Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| 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 | #16249 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3143 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)(13,14,16,17)(15,19,20,18)(22,25)(26,28);(6,10)(8,11)(13,14)(15,19)(16,17)(18,20);(6,10)(7,12)(14,17)(18,19)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,25)(27,28);(5,8,9,11)(6,7,10,12)(13,15)(14,18)(16,20)(17,19)(21,23,24,27)(22,26,25,28)Polycyclic presentation:
2^9|a^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72327196, #72327208, #72327282, … |
| maximal normal subgroup of | #72327196, #72327208, #72327282, … |
| maximal quotient of | #72332688 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4223679);