Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#394 |
| prime radical | #60949 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
931 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,10)(4,5)(8,15)(9,13)(11,12)(17,18)(19,22)(20,23)(21,24)(25,29)(26,28)(27,30)(31,32);(1,2)(3,8)(4,13)(5,7)(6,9)(10,14)(11,16)(12,15)(17,19)(18,22)(20,25)(21,26)(23,29)(24,28)(27,31)(30,32);(1,3,4,11,6,12,5,10)(2,8,7,14,9,15,13,16)(22,28)(29,32);(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,29,28,32)Polycyclic presentation:
2^9|c^2=e,d^2=i,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 148252);