Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682596 |
| 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 | #37669 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(13,14,18,22)(15,24,26,20)(16,25,27,28)(17,23,19,21);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15,18,26)(14,20,22,24)(16,23,27,21)(17,28,19,25);(2,4)(13,16,19,20,18,27,17,24)(14,15,23,25,22,26,21,28);(6,10)(9,11)(13,17,18,19)(14,21,22,23)(15,25,26,28)(16,20,27,24);(2,4)(7,12)(9,11)(15,26)(16,27)(20,24)(25,28)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590417);