Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| 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×#121 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1836 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)(6,9)(7,11)(14,18)(16,23)(19,25)(20,24)(22,27)(26,28);(1,3,4,2)(5,6)(7,10)(8,9)(11,12);(13,14,17,20)(15,19,21,26)(16,24,23,18)(22,28,27,25);(6,9)(10,12)(13,15)(14,19)(16,22)(17,21)(18,25)(20,26)(23,27)(24,28);(5,7,8,11)(6,10,9,12)(15,21)(19,26)(22,27)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,g^c=gi,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6295784);