Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470922 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1377 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)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,28,31,23)(20,33,35,40)(21,25,36,27)(24,39,41,34)(29,44,46,48)(30,42,32,38)(37,47,43,45);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,29)(21,32)(22,31)(24,37)(25,42)(26,28)(27,38)(30,36)(33,48)(34,47)(35,46)(39,45)(40,44)(41,43);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,20,21,34,22,35,36,39)(18,24,27,33,26,41,25,40)(19,29,32,47,31,46,30,45)(23,37,38,48,28,43,42,44);(11,15)(13,16)(17,21,22,36)(18,25,26,27)(19,30,31,32)(20,34,35,39)(23,38,28,42)(24,40,41,33)(29,45,46,47)(37,48,43,44);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(19,31)(20,35)(23,28)(24,41)(30,32)(33,40)(34,39)(38,42)Polycyclic presentation:
2^9|a^2=i,c^2=fgi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6382330);