Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×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 | |
4439 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)(7,12)(8,10)(9,11)(13,14)(15,19)(16,17)(18,20)(22,24)(23,28);(5,7,9,8)(6,10,11,12)(13,15)(14,18)(16,19)(17,20)(21,22,25,24)(23,27,28,26);(1,2)(3,4)(21,23)(22,26)(24,27)(25,28);(1,2)(3,4)(5,8,9,7)(6,10,11,12)(13,16,17,14)(15,19,20,18)(21,24,25,22)(23,27,28,26);(5,6)(7,10)(8,12)(9,11)(13,15)(14,19)(16,18)(17,20)(21,23)(22,27)(24,26)(25,28)Polycyclic presentation:
2^9|b^2=fgi,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72231231, #72232380, #72232426, … |
| maximal normal subgroup of | #72231231, #72232380, #72232426, … |
| maximal quotient of | #72278106, #72278341, #72293928, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4330337);