Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470402 |
| 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 | |
1684 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)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,28,33,46)(20,25,35,43)(21,26,22,37)(24,31,40,32)(29,44,30,45)(34,41,36,42)(38,48,39,47);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,32,23,33,21,31)(18,24,26,28,27,40,37,46)(20,29,34,47,35,30,36,48)(25,38,42,44,43,39,41,45);(1,3,4,8)(2,6,5,7)(11,15)(13,16)(17,20)(18,25)(19,29)(21,36)(22,34)(23,35)(24,38)(26,42)(27,43)(28,44)(30,33)(31,48)(32,47)(37,41)(39,40)(45,46);(11,15)(13,16)(17,20)(18,25)(19,30)(21,36)(22,34)(23,35)(24,39)(26,42)(27,43)(28,45)(29,33)(31,47)(32,48)(37,41)(38,40)(44,46);(10,14)(13,16)(17,21,23,22)(18,26,27,37)(19,31,33,32)(20,34,35,36)(24,28,40,46)(25,41,43,42)(29,47,30,48)(38,45,39,44)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=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, 6382717);