Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
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 | |
8487 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)(13,14)(15,21)(16,20)(17,19)(18,22)(23,27)(24,26)(25,28);(2,4)(5,6,8,10)(7,11,12,9);(5,7)(6,9)(8,12)(10,11)(13,15)(14,18)(16,23)(17,22)(19,21)(20,25)(24,28)(26,27);(6,10)(7,12)(15,22)(18,21)(23,28)(25,27);(7,12)(9,11)(14,19)(18,21)(20,26)(25,27);(13,16,17,24)(14,20,19,26)(15,23,22,28)(18,25,21,27)Polycyclic presentation:
2^9|b^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72280721, #72280763, #72282980, … |
| maximal normal subgroup of | #72280721, #72280763, #72282980, … |
| maximal quotient of | #72281199, #72304125 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7612858);