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 | #45423 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(7,11)(10,12)(13,14,19,21)(15,18,23,17)(16,20,25,28)(22,24,27,26);(2,3)(6,9)(7,11);(5,6)(7,10)(8,9)(11,12)(13,15,18,21,19,23,17,14)(16,22,24,28,25,27,26,20);(1,3,4,2)(6,9)(10,12)(13,16)(14,20)(15,22)(17,26)(18,24)(19,25)(21,28)(23,27);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,15,21,23)(16,24,25,26)(20,27,28,22)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=f,e^2=hi,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, 7063463);