Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| 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 | #23923 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1952 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)(17,18,23,28)(19,29,32,40)(20,26,34,25)(21,27,22,36)(24,37,41,31)(30,45,48,44)(33,43,35,42)(38,47,39,46);(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,30)(21,31)(22,37)(23,32)(25,38)(26,39)(27,40)(28,41)(29,36)(33,46)(34,48)(35,47)(42,44)(43,45);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(17,20,23,34)(18,25,28,26)(19,30,32,48)(21,35,22,33)(24,38,41,39)(27,43,36,42)(29,44,40,45)(31,47,37,46);(11,15)(13,16)(17,20)(18,26)(19,30)(21,35)(22,33)(23,34)(24,39)(25,28)(27,42)(29,45)(31,47)(32,48)(36,43)(37,46)(38,41)(40,44);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,21,23,22)(18,27,28,36)(19,31,32,37)(20,33,34,35)(24,40,41,29)(25,42,26,43)(30,46,48,47)(38,44,39,45)Polycyclic presentation:
2^9|a^2=i,c^2=g,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,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6383972);