Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #249019 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #36706 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(10,11)(13,14,17,20)(15,21,22,28)(16,24,25,19)(18,26,27,23);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(16,25)(19,24)(21,28)(23,26);(2,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,26)(17,22)(19,28)(20,27)(21,24)(23,25);(5,8,9,6)(7,11,12,10)(16,25)(19,24)(21,28)(23,26)Polycyclic presentation:
2^9|a^2=i,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=dhi,f^b=fh,e^c=eh,f^c=fi,f^d=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 156264);