Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,18,21,23)(19,27,30,42)(20,25,32,37)(22,35,28,26)(24,38,36,31)(29,43,33,46)(34,48,44,41)(39,47,40,45);(1,3,5,8)(2,6,4,7)(9,11,14,10)(12,15,13,16)(18,23)(20,32)(24,36)(26,35)(27,42)(29,33)(34,44)(45,47);(9,12)(10,15)(11,16)(13,14)(17,19,28,38,21,30,22,31)(18,24,35,42,23,36,26,27)(20,33,44,45,32,29,34,47)(25,40,48,43,37,39,41,46);(1,4,5,2)(3,6,8,7)(9,13,14,12)(10,16,11,15)(17,20,21,32)(18,25,23,37)(19,29,30,33)(22,34,28,44)(24,39,36,40)(26,41,35,48)(27,43,42,46)(31,45,38,47);(10,11)(15,16)(20,32)(25,37)(29,33)(34,44)(39,40)(41,48)(43,46)(45,47)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,c^2=gi,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6950627);