Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| 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×#3265 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2369 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(15,23)(16,17)(19,26)(20,22)(24,25)(27,28);(6,9)(7,11)(13,14,18,21)(15,19,24,27)(16,20,17,22)(23,26,25,28);(13,15,18,24)(14,19,21,27)(16,23,17,25)(20,26,22,28);(5,7,8,11)(6,10,9,12)(13,16,18,17)(14,20,21,22)(15,23,24,25)(19,26,27,28);(1,3,4,2)(6,9)(10,12)(14,21)(19,27)(20,22)(26,28)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,d^b=df,e^b=ei,g^c=gi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6306988);