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) = C25×C42 |
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 | |
3344 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)(7,11)(8,12)(9,10)(13,14,16,19)(15,18,20,17)(21,22,25,23)(24,26,28,27);(1,2,4,3)(5,7)(6,8)(9,12)(10,11)(13,15)(14,17)(16,20)(18,19)(22,23)(26,27);(2,3)(6,10)(8,11)(15,20)(17,18)(21,22,25,23)(24,27,28,26);(5,7)(6,8)(9,12)(10,11)(13,14)(15,18)(16,19)(17,20)(21,23,25,22)(24,27,28,26);(1,3,4,2)(5,8)(6,7)(9,11)(10,12)(13,15)(14,18)(16,20)(17,19)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=ghi,d^2=ghi,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 | #72326612, #72326616, #72326651, … |
| maximal normal subgroup of | #72326612, #72326616, #72326651, … |
| maximal quotient of | #72328881, #72328976, #72332982 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2322316);