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 | #2431 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(10,11)(12,16)(17,18,24,41,23,28,22,29)(19,30,39,25,33,47,34,44)(20,35,38,45,32,26,37,43)(21,27,36,48,40,42,31,46);(1,3)(2,5)(4,8)(6,7)(9,10)(11,13)(12,15)(14,16)(17,19)(18,25)(20,36)(21,38)(22,39)(23,33)(24,34)(26,27)(28,44)(29,47)(30,41)(31,32)(35,42)(37,40)(43,46)(45,48);(2,6)(5,7)(9,11,13,10)(12,15,16,14)(17,20)(18,26)(19,31)(21,39)(22,37)(23,32)(24,38)(25,42)(27,44)(28,35)(29,45)(30,48)(33,36)(34,40)(41,43)(46,47);(3,8)(5,7)(10,11)(14,15)(17,21,24,36,23,40,22,31)(18,27,41,48,28,42,29,46)(19,32,39,37,33,20,34,38)(25,43,47,35,44,45,30,26);(9,12,13,16)(10,14,11,15)(18,28)(19,33)(20,32)(27,42)(29,41)(34,39)(37,38)(46,48)Polycyclic presentation:
2^9|a^2=fh,c^2=hi,d^2=fg,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104711);