Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×#120 |
| 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×#127 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,12)(9,10);(6,10)(11,12)(13,14,18,15)(16,23,26,24)(17,19,27,21)(20,28,22,25);(1,2,3,4)(5,7,9,8)(6,11,10,12)(14,19)(15,21)(17,27)(20,23)(22,24)(25,28);(5,7,9,8)(6,11,10,12)(13,15,18,14)(16,24,26,23)(17,21,27,19)(20,25,22,28);(5,8,9,7)(6,12,10,11)(13,16,18,26)(14,20,15,22)(17,28,27,25)(19,23,21,24)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6492731);