Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234309 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(10,13)(11,15)(14,16)(17,18,21,24)(19,22,26,30)(20,23,28,32)(25,29,27,31);(2,5)(3,7)(17,19,21,26)(18,22,24,30)(20,27,28,25)(23,31,32,29);(9,10,11,13,12,14,15,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,32)(26,27)(30,31);(1,3,4,7)(2,6,5,8)(20,28)(23,32)(25,27)(29,31);(2,5)(6,8)(19,26)(22,30)(25,27)(29,31)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7174000);