Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469995 |
| 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 | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1937 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
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)(19,28)(20,26)(21,27)(22,35)(23,25)(24,30)(29,43)(31,39)(32,41)(33,40)(34,42)(36,37)(38,45)(44,47)(46,48);(9,11)(10,13)(12,15)(14,16)(17,19,23,31)(18,24,25,37)(20,29,33,46)(21,30,22,36)(26,38,40,47)(27,39,35,28)(32,44,34,45)(41,43,42,48);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(18,25)(24,37)(26,40)(27,35)(28,39)(38,47)(41,42)(43,48);(11,15)(13,16)(17,20)(18,26)(19,29)(21,34)(22,32)(23,33)(24,38)(25,40)(27,42)(28,43)(30,45)(31,46)(35,41)(36,44)(37,47)(39,48);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,21,23,22)(18,27,25,35)(19,30,31,36)(20,32,33,34)(24,39,37,28)(26,41,40,42)(29,44,46,45)(38,43,47,48)Polycyclic presentation:
2^9|b^2=i,c^2=gi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6383932);