Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3397 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,12)(9,11)(13,14,18,22)(15,19,17,20)(16,21,24,28)(23,26,25,27);(5,7,10,6)(8,12,9,11)(13,15,18,17)(14,19,22,20)(16,23,24,25)(21,26,28,27);(5,8,10,9)(6,11,7,12)(14,20)(15,17)(19,22)(21,27)(23,25)(26,28);(5,8,10,9)(6,12,7,11)(13,16)(14,21)(15,23)(17,25)(18,24)(19,26)(20,27)(22,28);(1,3,4,2)(5,9,10,8)(6,12,7,11)(16,24)(21,28)(23,25)(26,27)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=f,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,g^c=gi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6308205);