Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
4239 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,7)(10,11)(13,14)(15,16)(17,19)(18,20)(22,25)(23,27);(2,4)(5,6,9,7)(8,11,12,10)(13,15,17,20)(14,18,19,16)(21,22,24,25)(23,28,27,26);(1,2)(3,4)(6,7)(10,11)(14,19)(16,18)(21,22,24,25)(23,26,27,28);(5,7,9,6)(8,11,12,10)(13,16,17,18)(14,15,19,20)(21,22)(23,26)(24,25)(27,28);(5,8)(6,10)(7,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,23)(22,26)(24,27)(25,28)Polycyclic presentation:
2^9|b^2=f,c^2=fgi,d^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232324, #72232340, #72232394, … |
| maximal normal subgroup of | #72232324, #72232340, #72232394, … |
| maximal quotient of | #72323649, #72328869, #72328929, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1426396);