Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C2×#127 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C42×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1478 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)(3,7)(4,6)(5,8)(17,18,23,28)(19,25,30,31)(20,24,29,32)(21,26,22,27);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,13)(14,16)(17,19,21,20,23,30,22,29)(18,24,27,25,28,32,26,31);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,20,21,30,23,29,22,19)(18,25,27,32,28,31,26,24);(9,11,12,16)(10,14,13,15)(17,21,23,22)(18,26,28,27)(19,29,30,20)(24,25,32,31);(1,4,5,3)(2,6,8,7)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,d^2=h,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372480);