Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| 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 | #36258 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2416 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)(8,10)(9,12);(5,7)(6,9)(8,12)(10,11)(13,14,17,21)(15,22,24,19)(16,18,25,20)(23,27,28,26);(1,2,3,4)(14,18)(16,25)(19,26)(20,21)(22,27)(23,28);(6,10)(9,11)(13,15,17,24)(14,19,21,22)(16,23,25,28)(18,26,20,27);(7,12)(9,11)(14,18)(16,25)(19,26)(20,21)(22,27)(23,28)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72283018, #72283639, #72284010, … |
| maximal normal subgroup of | #72283018, #72283639, #72284010, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589422);