Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324500 |
| 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 | |
1722 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)(17,18)(19,29)(20,26)(21,28)(22,24)(23,27)(25,31)(30,43)(32,41)(33,38)(34,42)(35,37)(36,40)(39,46)(44,48)(45,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(18,24)(20,33)(25,37)(27,28)(29,41)(30,44)(34,36)(45,46);(2,6)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,23,31,22,32,21,35)(18,25,28,29,24,37,27,41)(20,30,34,46,33,44,36,45)(26,39,40,43,38,47,42,48);(3,8)(5,7)(10,14)(11,16)(17,20)(18,26)(19,30)(21,36)(22,33)(23,34)(24,38)(25,39)(27,42)(28,40)(29,43)(31,46)(32,44)(35,45)(37,47)(41,48);(2,6)(5,7)(10,14)(13,15)(17,21,22,23)(18,27,24,28)(19,31,32,35)(20,34,33,36)(25,29,37,41)(26,40,38,42)(30,45,44,46)(39,48,47,43)Polycyclic presentation:
2^9|a^2=fi,c^2=fgi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7234149);