Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| 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 | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(13,14,19,23)(15,24,26,16)(17,21,25,28)(18,22,20,27);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,19,26)(14,16,23,24)(17,20,25,18)(21,22,28,27);(2,4)(13,16,18,21,19,24,20,28)(14,17,27,26,23,25,22,15);(2,4)(5,8,9,7)(6,10,11,12)(13,17,19,25)(14,21,23,28)(15,18,26,20)(16,27,24,22);(13,18,19,20)(14,22,23,27)(15,25,26,17)(16,21,24,28)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351124);