Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200022 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2264 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,30)(21,26)(22,35)(23,27)(25,28)(29,38)(31,43)(32,37)(33,41)(34,45)(36,40)(39,42)(44,47)(46,48);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,22)(18,24,27,35)(20,28,32,36)(21,29,33,34)(25,37,40,30)(26,38,41,45)(31,42,44,46)(39,47,48,43);(11,15)(13,16)(17,20,23,32)(18,25,27,40)(19,28,22,36)(21,31,33,44)(24,37,35,30)(26,39,41,48)(29,42,34,46)(38,47,45,43);(3,8)(5,7)(10,14)(11,15)(17,21)(18,26)(19,29)(20,31)(22,34)(23,33)(24,38)(25,39)(27,41)(28,42)(30,43)(32,44)(35,45)(36,46)(37,47)(40,48);(2,6)(5,7)(11,15)(13,16)(21,33)(26,41)(29,34)(31,44)(38,45)(39,48)(42,46)(43,47)Polycyclic presentation:
2^9|b^2=i,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7107228);