Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6)(7,10)(8,13)(9,14)(11,15)(12,18)(16,19)(17,20)(21,22,25,26)(23,29,31,27)(24,28,33,32)(30,35,36,34)(37,38)(39,43)(40,42)(41,48)(44,47)(45,49)(46,51)(50,52);(6,7)(8,15)(12,14)(16,17)(22,26)(27,29)(28,32)(34,35)(37,39,42,47)(38,43,40,44)(41,46,49,52)(45,50,48,51);(5,7,10,6)(8,13,15,11)(9,14,18,12)(16,20,17,19)(21,23,25,31)(22,27,26,29)(24,30,33,36)(28,34,32,35)(37,39)(38,44)(40,43)(41,46)(42,47)(45,51)(48,50)(49,52);(1,2)(3,4)(5,8)(6,11)(7,13)(9,17)(10,15)(12,20)(14,19)(16,18)(21,22)(23,27)(24,32)(25,26)(28,33)(29,31)(30,35)(34,36)(37,40,42,38)(39,43,47,44)(41,48,49,45)(46,50,52,51);(5,9,10,18)(6,12,7,14)(8,16,15,17)(11,19,13,20)(21,24)(22,28)(23,30)(25,33)(26,32)(27,34)(29,35)(31,36)(37,41,42,49)(38,45,40,48)(39,46,47,52)(43,50,44,51)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=f,d^2=fg,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4772706);