Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573511 |
| 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 |
| prime radical | #25763 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,12)(9,10);(6,10)(11,12)(13,14,18,22)(15,20,24,27)(16,21,17,19)(23,28,25,26);(1,2,3,4)(5,7,9,8)(6,11,10,12)(14,19)(15,23)(16,17)(20,27)(21,22)(24,25);(5,8,9,7)(6,12,10,11)(13,15,18,24)(14,20,22,27)(16,23,17,25)(19,26,21,28);(13,16)(17,18)(19,21)(20,28)(23,25)(26,27)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72282258, #72283022, #72283201, … |
| maximal normal subgroup of | #72282258, #72283022, #72283201, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6491722);