Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
3489 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,8,10)(7,11,12,9)(13,14)(15,18)(16,19)(17,20)(21,22)(23,25)(24,28)(26,27);(1,2,4,3)(5,7)(6,9)(8,12)(10,11)(13,15)(14,17)(16,20)(18,19)(22,27)(23,24);(2,3)(6,10)(9,11)(15,20)(17,18)(21,23,26,24)(22,25,27,28);(5,7)(6,9)(8,12)(10,11)(13,14)(15,18)(16,19)(17,20)(21,24,26,23)(22,25,27,28);(1,3,4,2)(5,6)(7,9)(8,10)(11,12)(13,15)(14,18)(16,20)(17,19)(21,25)(22,23)(24,27)(26,28)Polycyclic presentation:
2^9|a^2=fh,b^2=fgh,c^2=fhi,d^2=fhi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232882, #72232901, #72232945, … |
| maximal normal subgroup of | #72232882, #72232901, #72232945, … |
| maximal quotient of | #72328742, #72328881, #72330635 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2322936);