Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | |
1329 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)(27,38)(28,41)(29,47)(30,48)(31,37)(32,35)(33,40)(34,45)(36,49)(39,50)(42,51)(43,52)(44,53)(46,56)(54,55);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27)(26,33)(28,42)(29,30)(31,40)(32,49)(34,51)(35,48)(36,47)(37,38)(39,46)(41,53)(43,55)(44,45)(50,52)(54,56);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,32,46,31,45,29,43)(26,34,35,52,37,41,47,56)(27,39,30,42,40,55,49,44)(33,50,36,51,38,54,48,53);(10,16)(14,17)(15,22)(21,23)(25,29,31,32)(26,35,37,47)(27,30,40,49)(28,43,45,46)(33,48,38,36)(34,52,41,56)(39,42,55,44)(50,53,54,51);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(21,23)(25,30,32,40,31,49,29,27)(26,36,35,38,37,48,47,33)(28,44,46,39,45,42,43,55)(34,53,52,50,41,51,56,54)Polycyclic presentation:
2^9|b^2=fi,c^2=fgh,d^2=i,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,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, 6743132);