Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 |
| prime radical | C22×#553 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3540 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)(8,11)(9,12)(14,19)(15,23)(16,18)(20,27)(21,22)(24,28);(2,4)(5,7,9,8)(6,10,12,11)(13,14,17,21)(15,24,23,27)(16,22,18,19)(20,25,28,26);(5,7,9,8)(6,11,12,10)(13,15,17,23)(14,20,21,28)(16,26,18,25)(19,24,22,27);(5,8,9,7)(6,11,12,10)(14,19)(16,18)(20,24)(21,22)(25,26)(27,28);(14,21)(19,22)(20,28)(24,27)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72257755, #72257762, #72282404, … |
| maximal normal subgroup of | #72257755, #72257762, #72282404, … |
| maximal quotient of | #72257812, #72316396, #72316506 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6360196);