Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994107 |
| 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 | #59017 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1922 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)(7,9)(8,10)(11,12)(13,14,18,26,17,20,16,21)(15,22,25,19,23,28,24,27);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(15,23)(19,27)(22,28)(24,25);(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,25)(17,23)(18,24)(20,27)(21,28)(22,26);(2,4)(7,12)(9,11)(14,20)(15,23)(21,26)(24,25);(6,10)(9,11)(14,20)(19,27)(21,26)(22,28)Polycyclic presentation:
2^9|a^2=gi,c^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6969599);