Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432861 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1704 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,29,31,37)(20,26,34,41)(21,27,22,25)(24,33,39,32)(30,43,46,47)(35,42,36,40)(38,45,48,44);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,31)(18,24,28,39)(20,30,34,46)(21,33,22,32)(25,37,27,29)(26,38,41,48)(35,45,36,44)(40,47,42,43);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(18,25)(21,22)(24,37)(26,40)(27,28)(29,39)(32,33)(35,36)(38,47)(41,42)(43,48)(44,45);(10,14)(13,16)(17,20)(18,26)(19,30)(21,35)(22,36)(23,34)(24,38)(25,40)(27,42)(28,41)(29,43)(31,46)(32,44)(33,45)(37,47)(39,48);(19,31)(20,34)(24,39)(26,41)(29,37)(32,33)(35,36)(40,42)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,c^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341316);