Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470884 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2125 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)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,20,17)(15,22,27,28)(16,21,26,25)(18,23,19,24);(5,7)(6,9)(8,11)(10,12)(13,15,19,26,20,27,18,16)(14,21,23,28,17,25,24,22);(2,4)(7,11)(9,12)(13,16,18,27,20,26,19,15)(14,22,24,25,17,28,23,21);(7,11)(9,12)(13,17,20,14)(15,25,27,21)(16,28,26,22)(18,23,19,24);(6,10)(9,12)(13,18,20,19)(14,23,17,24)(15,16,27,26)(21,28,25,22)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6378139);