Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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)(27,31)(28,39)(29,33)(30,34)(32,41)(35,53)(36,51)(37,47)(38,46)(40,56)(42,52)(43,50)(44,49)(45,54)(48,55);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27,29,37)(26,31,33,47)(28,40,41,55)(30,36,43,38)(32,48,39,56)(34,51,50,46)(35,44,54,42)(45,52,53,49);(2,5)(6,8)(25,28,29,41)(26,32,33,39)(27,35,37,54)(30,44,43,42)(31,45,47,53)(34,52,50,49)(36,40,38,55)(46,56,51,48);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,23)(22,24)(28,41)(32,39)(35,54)(40,55)(42,44)(45,53)(48,56)(49,52);(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(26,33)(27,36)(28,41)(30,43)(31,46)(35,55)(37,38)(40,54)(45,48)(47,51)(49,52)(53,56)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6611630);