Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533105 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3511 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3146 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)(13,14)(15,25)(16,27)(17,24)(18,22)(19,23)(20,26)(21,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,16)(14,20,23,21)(17,26,18,28)(22,25,24,27);(6,9)(10,12)(13,16,19,15)(14,21,23,20)(17,28,18,26)(22,27,24,25);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,23,24)(15,26,16,28)(20,25,21,27);(7,11)(10,12)(14,23)(20,21)(22,24)(25,27)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=i,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,g^a=gi,c^b=ch,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6449479);