Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #45422 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(7,11)(10,12)(14,20)(15,22)(17,19)(21,24)(23,26)(25,28);(2,3)(6,9)(7,11)(13,14)(15,23)(16,19)(17,27)(18,20)(21,25)(22,24)(26,28);(5,6)(7,10)(8,9)(11,12)(13,15,19,22,18,25,17,28)(14,21,16,26,20,23,27,24);(1,3,4,2)(6,9)(10,12)(13,16)(14,17)(15,24)(18,27)(19,20)(21,22)(23,28)(25,26);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,16,20,27)(15,22,25,28)(21,24,23,26)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7063461);