Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535815 |
| 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 | #25384 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,4,8)(9,10,13,11)(12,14,16,15)(17,18,22,27)(19,28,31,20)(21,25,29,26)(23,24,32,30);(2,6)(7,8)(9,11,13,10)(12,15,16,14)(17,19,29,24,22,31,21,30)(18,23,25,20,27,32,26,28);(1,3,5,4)(2,7,6,8)(10,11)(14,15)(17,20,21,23,22,28,29,32)(18,24,26,19,27,30,25,31);(1,4,5,3)(2,8,6,7)(12,16)(14,15)(17,21,22,29)(18,25,27,26)(19,30,31,24)(20,32,28,23);(9,12,13,16)(10,14,11,15)(17,21,22,29)(18,25,27,26)(19,24,31,30)(20,23,28,32)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,c^2=f,d^2=fgh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6465142);