Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433057 |
| 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 | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2090 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)(13,14,19,23)(15,24,26,28)(16,25,22,18)(17,27,20,21);(5,7)(6,9)(8,11)(10,12)(13,15,25,21,19,26,18,27)(14,20,16,28,23,17,22,24);(2,4)(7,11)(9,12)(13,16,18,14,19,22,25,23)(15,24,27,17,26,28,21,20);(6,10)(9,12)(13,17)(14,21)(15,22)(16,26)(18,28)(19,20)(23,27)(24,25);(15,26)(17,20)(21,27)(24,28)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^b=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340486);