Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,12)(7,8,11,10)(13,14)(15,18)(16,24)(17,26)(19,23)(20,25)(21,28)(22,27);(1,2)(3,4)(5,7,9,11)(6,10,12,8)(13,15,18,14)(16,22,17,20)(19,23,28,21)(24,25,26,27);(2,4)(5,8,9,10)(6,11,12,7)(13,16,18,17)(14,20,15,22)(19,27,28,25)(21,26,23,24);(6,12)(7,11)(13,17,18,16)(14,20,15,22)(19,25,28,27)(21,26,23,24);(14,15)(20,22)(21,23)(24,26)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=f,d^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,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, 7409193);