Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62283 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1631 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,4,3)(5,6)(7,15)(8,12)(9,13)(10,14)(11,17)(16,20)(18,19)(21,22,25,27)(23,31,32,45)(24,29,35,40)(26,38,37,30)(28,41,39,34)(33,44,46,52)(36,49,48,43)(42,51,50,47);(5,7,10,17)(6,11,14,15)(8,16,9,18)(12,19,13,20)(22,27)(23,32)(24,35)(30,38)(34,41)(36,48)(42,50)(44,52);(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,23,26,34,25,32,37,41)(22,28,30,31,27,39,38,45)(24,33,36,47,35,46,48,51)(29,42,43,44,40,50,49,52);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(21,24)(22,29)(23,33)(25,35)(26,36)(27,40)(28,42)(30,43)(31,44)(32,46)(34,47)(37,48)(38,49)(39,50)(41,51)(45,52);(1,3,4,2)(6,14)(11,15)(12,13)(19,20)(22,27)(24,35)(28,39)(30,38)(31,45)(33,46)(36,48)(47,51)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=fgi,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6311694);