Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | #39131 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,7)(3,4,8,5)(9,10)(11,15)(12,14)(13,16)(17,18,21,23)(19,27,28,43)(20,32,34,25)(22,36,29,26)(24,38,37,31)(30,45,33,42)(35,41,44,47)(39,46,40,48);(1,3)(2,5)(4,7)(6,8)(9,11)(10,13)(12,16)(14,15)(18,23)(19,28)(25,32)(26,36)(30,33)(31,38)(41,47)(46,48);(1,3)(2,4)(5,7)(6,8)(11,16)(13,15)(17,19,29,38,21,28,22,31)(18,24,36,43,23,37,26,27)(20,33,44,46,34,30,35,48)(25,40,47,42,32,39,41,45);(1,4,6,5)(2,3,7,8)(10,14)(13,15)(17,20)(18,25)(19,30)(21,34)(22,35)(23,32)(24,39)(26,41)(27,42)(28,33)(29,44)(31,46)(36,47)(37,40)(38,48)(43,45);(1,5,6,4)(2,3,7,8)(20,34)(25,32)(30,33)(35,44)(39,40)(41,47)(42,45)(46,48)Polycyclic presentation:
2^9|a^2=fhi,c^2=gi,d^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfhi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6897203);