Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| 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 | |
2130 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)(8,9)(11,12)(15,22)(20,28)(23,24)(25,26);(2,4)(13,14,18,21)(15,23,22,24)(16,19,27,17)(20,25,28,26);(6,9)(10,12)(13,15,19,26,18,22,17,25)(14,20,16,23,21,28,27,24);(13,16,19,21,18,27,17,14)(15,24,26,20,22,23,25,28);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,16,21,27)(15,25,22,26)(20,23,28,24)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=h,e^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72249920, #72256930, #72256969, … |
| maximal normal subgroup of | #72249920, #72256930, #72256969, … |
| maximal quotient of | #72264296, #72264332, #72264355 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374502);