Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996607 |
| 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×#3308 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1920 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)(7,10)(8,11)(9,12)(13,14,17,21,18,20,25,27)(15,22,24,26,23,28,16,19);(5,6)(7,10)(8,11)(9,12)(13,15,18,23)(14,19,20,26)(16,25,24,17)(21,28,27,22);(1,2,3,4)(5,7,10,6)(8,11,9,12)(15,23)(16,24)(19,26)(22,28);(5,8,10,9)(6,11,7,12)(13,16,17,15,18,24,25,23)(14,19,21,22,20,26,27,28);(5,9,10,8)(6,11,7,12)(14,20)(15,23)(16,24)(21,27)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=fhi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7106277);