Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574024 |
| 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×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1829 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)(14,19)(15,23)(18,21)(20,27)(24,25)(26,28);(2,4)(5,6,8,9)(7,10,11,12)(13,14,17,19)(15,24,26,20)(16,18,22,21)(23,27,28,25);(6,9)(7,11)(13,15,21,28,17,26,18,23)(14,20,16,27,19,24,22,25);(5,7)(6,10)(8,11)(9,12)(13,16,17,22)(14,21,19,18)(15,25,26,27)(20,23,24,28);(6,9)(7,11)(14,19)(16,22)(20,24)(25,27)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=gi,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^d=eh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6513598);