Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,18)(16,20)(17,19)(21,24)(22,25)(23,26)(27,28);(2,4)(7,8)(10,11)(13,15)(14,18)(16,22)(17,21)(19,24)(20,25)(23,27)(26,28);(5,7,9,8)(6,10,12,11)(14,19)(18,24)(20,26)(25,28);(5,8,9,7)(6,11,12,10)(14,19)(15,21)(20,26)(22,27);(6,12)(10,11)(14,19)(18,24)(20,26)(25,28);(13,16,17,23)(14,20,19,26)(15,22,21,27)(18,25,24,28)Polycyclic presentation:
2^9|c^2=i,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,e^a=ehi,c^b=ci,d^b=dhi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72263912, #72263922, #72264014, … |
| maximal normal subgroup of | #72263912, #72263922, #72264014, … |
| maximal quotient of | #72264116, #72305136, #72305140 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541102);