Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432861 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1802 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(18,24)(19,27)(20,28)(21,22)(25,26)(29,30);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,18,23,25)(19,28,30,20)(21,26,22,24)(27,32,29,31);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,19)(18,20)(21,27)(22,29)(23,30)(24,32)(25,28)(26,31);(10,14)(13,16)(17,20)(18,19)(21,31)(22,32)(23,28)(24,29)(25,30)(26,27);(18,25)(20,28)(24,26)(31,32)Polycyclic presentation:
2^9|b^2=fgi,c^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341315);