Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3012 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2821 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,11,12,9)(15,22)(16,26)(17,18)(20,27)(23,28)(24,25);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,14,19,21)(15,23,25,16)(17,27,18,20)(22,28,24,26);(2,4)(13,15)(14,16)(17,22)(18,24)(19,25)(20,28)(21,23)(26,27);(6,10)(7,12)(13,16)(14,15)(17,26)(18,28)(19,23)(20,24)(21,25)(22,27);(13,17,19,18)(14,20,21,27)(15,24,25,22)(16,26,23,28)Polycyclic presentation:
2^9|a^2=f,b^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408324);