Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437521 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3038 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,22)(15,17,21,18)(16,20,27,24)(23,25,28,26);(1,2,3,4)(13,15,17,22,19,21,18,14)(16,24,26,23,27,20,25,28);(6,9)(10,12)(13,16)(14,20)(15,23)(17,25)(18,26)(19,27)(21,28)(22,24);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,23)(17,25)(18,26)(19,27)(21,28)(22,24);(13,17,19,18)(14,15,22,21)(16,25,27,26)(20,23,24,28)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^c=fi,h^c=hi,f^d=fi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6356414);