Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| 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 | #2541 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1631 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(17,18,23,19)(20,22,24,21);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,13)(14,16)(17,19,22,24,23,18,21,20);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,20,22,19,23,24,21,18);(9,11,12,16)(10,14,13,15)(17,21,23,22)(18,20,19,24);(1,4,5,3)(2,6,8,7)(17,21,23,22)(18,20,19,24)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,e^b=ei,d^c=di,e^c=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372576);