Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2394 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)(7,10)(8,12)(9,11)(13,14)(15,25)(16,27)(17,28)(18,22)(19,24)(20,23)(21,26);(5,7,10,6)(8,12,9,11)(13,15,19,26)(14,21,24,25)(16,20,18,17)(22,23,27,28);(2,4)(13,16,20,26,19,18,17,15)(14,22,23,25,24,27,28,21);(5,8,10,9)(6,11,7,12)(13,17,19,20)(14,23,24,28)(15,16,26,18)(21,27,25,22);(5,9,10,8)(6,11,7,12)(13,18,17,26,19,16,20,15)(14,22,28,21,24,27,23,25)Polycyclic presentation:
2^9|b^2=f,c^2=h,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72256854, #72256878, #72257047, … |
| maximal normal subgroup of | #72256854, #72256878, #72257047, … |
| maximal quotient of | #72264739, #72264863 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718755);