Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(15,22)(20,27)(23,26)(24,28);(2,4)(14,18)(15,23)(16,19)(20,27)(21,25)(22,26);(6,9)(10,12)(13,14,19,25,17,21,16,18)(15,20,26,28,22,27,23,24);(13,15,16,23,17,22,19,26)(14,20,18,24,21,27,25,28);(5,7,8,11)(6,10,9,12)(14,21)(18,25)(20,27)(24,28)Polycyclic presentation:
2^9|c^2=hi,d^2=h,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,h^b=hi,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72307249, #72307267, #72307624, … |
| maximal normal subgroup of | #72307249, #72307267, #72307624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374475);