Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470374 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1727 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,33)(21,26)(22,37)(23,27)(24,31)(25,35)(29,44)(30,39)(32,43)(34,41)(36,40)(38,47)(42,46)(45,48);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,30,23,32,22,31)(18,24,37,43,27,39,26,28)(20,29,36,47,35,46,34,45)(25,38,40,44,33,48,41,42);(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)(26,41)(27,33)(28,42)(30,47)(31,45)(32,46)(37,40)(39,48)(43,44);(11,15)(13,16)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,34,35,36)(24,28,39,43)(25,40,33,41)(29,45,46,47)(38,44,48,42);(10,14)(13,16)(20,35)(25,33)(29,46)(34,36)(38,48)(40,41)(42,44)(45,47)Polycyclic presentation:
2^9|b^2=fg,c^2=gi,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6382691);