Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | |
1339 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)(27,32)(28,33)(29,34)(30,36)(31,35)(37,45)(38,46)(39,51)(40,47)(41,48)(42,49)(43,50)(44,52)(53,55)(54,56);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27,31,40)(26,32,35,47)(28,41,43,53)(29,38,30,39)(33,48,50,55)(34,46,36,51)(37,44,54,42)(45,52,56,49);(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)(26,33)(27,37)(29,42)(30,44)(31,43)(32,45)(34,49)(35,50)(36,52)(38,53)(39,41)(40,54)(46,55)(47,56)(48,51);(10,16)(14,17)(15,22)(21,23)(25,29,31,30)(26,34,35,36)(27,38,40,39)(28,42,43,44)(32,46,47,51)(33,49,50,52)(37,53,54,41)(45,55,56,48);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(21,23)(26,35)(27,39)(29,30)(32,46)(33,50)(37,41)(38,40)(42,44)(45,55)(47,51)(48,56)(53,54)Polycyclic presentation:
2^9|b^2=fi,c^2=fg,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6629130);