Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #45245 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1664 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,29,30,41)(20,25,34,40)(21,27,22,26)(24,35,37,32)(31,43,44,48)(33,42,36,39)(38,45,47,46);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(19,30)(24,37)(29,41)(31,44)(32,35)(38,47)(43,48)(45,46);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,32,23,30,21,35)(18,24,27,29,28,37,26,41)(20,31,33,46,34,44,36,45)(25,38,39,43,40,47,42,48);(2,5)(6,8)(11,16)(13,15)(17,20)(18,25)(19,31)(21,36)(22,33)(23,34)(24,38)(26,42)(27,39)(28,40)(29,43)(30,44)(32,46)(35,45)(37,47)(41,48);(10,14)(13,15)(17,21,23,22)(18,26,28,27)(19,32,30,35)(20,33,34,36)(24,29,37,41)(25,39,40,42)(31,45,44,46)(38,48,47,43)Polycyclic presentation:
2^9|a^2=i,c^2=fg,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6926996);