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 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1217 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)(3,6)(4,5)(7,8)(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26)(27,36)(28,33)(29,43)(30,35)(31,34)(32,39)(37,53)(38,47)(40,56)(41,52)(42,51)(44,48)(45,50)(46,55)(49,54);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27)(26,32)(28,40)(29,38)(30,39)(31,44)(33,49)(34,47)(35,36)(37,41)(42,54)(43,48)(45,55)(46,52)(50,53)(51,56);(2,5)(6,8)(25,28,30,42)(26,33,35,51)(27,37,39,55)(29,41,31,45)(32,46,36,53)(34,50,43,52)(38,54,44,40)(47,49,48,56);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,23)(22,24)(25,29,30,31)(26,34,35,43)(27,38,39,44)(28,41,42,45)(32,47,36,48)(33,50,51,52)(37,54,55,40)(46,49,53,56);(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(26,35)(27,38)(29,31)(32,47)(33,51)(36,48)(37,54)(39,44)(40,55)(41,45)(46,49)(53,56)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=gi,e^2=fi,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, 6611343);