Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682596 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2639 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,9)(7,11)(13,14,18,21)(15,23,25,20)(16,17,22,19)(24,26,28,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,28)(17,26)(18,25)(19,27)(21,23)(22,24);(2,4)(13,16,19,14,18,22,17,21)(15,24,27,23,25,28,26,20);(5,7,8,11)(6,10,9,12)(14,21)(15,25)(16,22)(26,27);(2,4)(6,9)(10,12)(13,17,18,19)(14,22,21,16)(15,26,25,27)(20,24,23,28)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72282961, #72283011, #72283539, … |
| maximal normal subgroup of | #72282961, #72283011, #72283539, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590919);