Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500186 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ 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)(3,5)(4,6)(7,8)(25,26,30,35,31,36,29,34)(27,37,41,46,38,48,40,32)(28,42,44,50,43,33,45,49)(39,53,54,47,52,51,55,56);(1,3,4,7)(2,5,6,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27)(26,32)(28,39)(29,41)(30,40)(31,38)(33,47)(34,37)(35,48)(36,46)(42,56)(43,52)(44,55)(45,54)(49,53)(50,51);(2,6)(5,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(27,38)(28,43)(32,46)(33,42)(37,48)(40,41)(44,45)(49,50);(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,33)(27,39)(29,45)(30,44)(31,43)(32,47)(34,50)(35,49)(36,42)(37,51)(38,52)(40,55)(41,54)(46,56)(48,53);(10,16)(14,17)(15,24)(22,23)(25,29,31,30)(26,34,36,35)(27,40,38,41)(28,44,43,45)(32,48,46,37)(33,49,42,50)(39,54,52,55)(47,51,56,53)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=hi,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6439362);