Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| 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 | #35564 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2063 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)(5,6)(7,9)(8,10)(11,12)(13,14,17,18)(15,19,22,25)(16,23,24,20)(21,27,28,26);(5,7,9,6)(8,11,12,10)(14,18)(15,19)(16,24)(21,27)(22,25)(26,28);(2,4)(5,8,9,12)(6,10,7,11)(13,15,17,22)(14,19,18,25)(16,21,24,28)(20,26,23,27);(2,4)(6,7)(10,11)(13,16)(14,20)(15,21)(17,24)(18,23)(19,26)(22,28)(25,27);(16,24)(20,23)(21,28)(26,27)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410886);