Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #23854 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,30)(19,25,31,20)(21,27,26,32)(22,29,23,28);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19,24,31)(18,25,30,20)(21,22,26,23)(27,29,32,28);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,20,23,27,24,25,22,32)(18,26,29,19,30,21,28,31);(2,6)(7,8)(11,16)(13,15)(17,21)(18,27)(19,22)(20,28)(23,31)(24,26)(25,29)(30,32);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,21,31,26)(20,27,25,32)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6934596);