Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,36)(27,32,39,46)(28,33,37,44)(29,34,40,47)(30,35,43,50)(38,45,52,55)(41,48,51,54)(42,49,53,56);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27)(26,32)(28,40)(29,37)(30,41)(31,39)(33,47)(34,44)(35,48)(36,46)(38,42)(43,51)(45,49)(50,54)(52,53)(55,56);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,40,27,31,37,29,39)(26,33,47,32,36,44,34,46)(30,41,53,38,43,51,42,52)(35,48,56,45,50,54,49,55);(10,16)(14,17)(15,22)(21,23)(25,29,31,40)(26,34,36,47)(27,37,39,28)(30,42,43,53)(32,44,46,33)(35,49,50,56)(38,51,52,41)(45,54,55,48);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(21,23)(25,30,29,42,31,43,40,53)(26,35,34,49,36,50,47,56)(27,38,28,41,39,52,37,51)(32,45,33,48,46,55,44,54)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fghi,d^2=i,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6743175);