Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,12)(8,11)(9,10)(13,14)(15,18)(16,20)(17,19)(22,25)(23,27);(1,3,4,2)(6,10)(11,12)(13,15,17,16)(14,18,19,20)(21,22)(23,26)(24,25)(27,28);(13,16,17,15)(14,18,19,20)(22,25)(26,28);(5,7,9,8)(6,11,10,12)(21,23,24,27)(22,26,25,28);(5,8,9,7)(6,11,10,12)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=g,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 421700);