Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | #42876 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1918 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)(6,9)(7,11)(13,14,18,22)(15,17,23,20)(16,21,25,27)(19,24,26,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,23)(14,20,22,17)(16,26,25,19)(21,24,27,28);(2,4)(13,16,19,15,18,25,26,23)(14,21,24,17,22,27,28,20);(5,7,8,11)(6,10,9,12)(14,22)(17,20)(21,27)(24,28);(6,9)(10,12)(13,17,26,21,18,20,19,27)(14,23,28,25,22,15,24,16)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=fi,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715390);