Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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 | #23863 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ 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:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,23)(16,19)(17,21)(18,26)(20,22)(24,28)(25,27);(5,7,9,8)(6,10,12,11)(13,15)(14,21)(16,23)(17,19)(18,25)(20,28)(22,27)(24,26);(2,4)(5,7,9,8)(6,11,12,10)(13,16,19,14)(15,24,17,27)(18,26,20,22)(21,28,23,25);(5,8,9,7)(6,10,12,11)(13,17)(14,21)(15,19)(16,23)(18,28)(20,25)(22,27)(24,26);(13,18)(14,22)(16,26)(19,20)(21,23)(25,28)Polycyclic presentation:
2^9|b^2=fi,c^2=f,d^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72263948, #72263990, #72264013 |
| maximal normal subgroup of | #72263948, #72263990, #72264013 |
| maximal quotient of | #72265299, #72308314 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6484344);