Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
3402 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)(13,14)(15,24)(16,27)(17,22)(18,23)(19,21)(20,26)(25,28);(2,4)(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,21)(17,27)(18,26)(19,25)(22,24)(23,28);(6,9)(7,11)(13,16,19,20)(14,15,21,25)(17,24,18,28)(22,26,23,27);(5,7)(6,10)(8,11)(9,12)(14,21)(15,25)(22,23)(26,27);(6,9)(7,11)(13,17,19,18)(14,22,21,23)(15,26,25,27)(16,24,20,28)Polycyclic presentation:
2^9|b^2=h,c^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72305353, #72305363, #72305374 |
| maximal normal subgroup of | #72305353, #72305363, #72305374 |
| maximal quotient of | #72305611, #72305615, #72305628, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503233);