Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1554 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)(9,12)(11,15)(14,17)(16,19)(18,20)(21,22,26,27)(23,24,29,32)(25,28,34,35)(30,31,33,36)(37,38,41,44)(39,45,47,42)(40,48,49,43)(46,50,52,51);(6,7)(8,15)(12,14)(16,20)(22,27)(23,29)(28,35)(30,33)(37,38,41,44)(39,45,47,42)(40,43,49,48)(46,51,52,50);(5,7,10,6)(8,13,15,11)(9,12,17,14)(16,18,20,19)(21,23)(22,24)(25,33)(26,29)(27,32)(28,36)(30,34)(31,35)(37,38)(39,42)(40,43)(41,44)(45,47)(46,50)(48,49)(51,52);(1,2)(3,4)(5,8)(6,11)(7,13)(9,16)(10,15)(12,18)(14,19)(17,20)(21,24)(22,23)(25,31)(26,32)(27,29)(28,30)(33,35)(34,36)(37,39,41,47)(38,42,44,45)(40,46,49,52)(43,50,48,51);(5,9,10,17)(6,12,7,14)(8,16,15,20)(11,18,13,19)(21,25,26,34)(22,28,27,35)(23,30,29,33)(24,31,32,36)(37,40)(38,43)(39,46)(41,49)(42,50)(44,48)(45,51)(47,52)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=fgh,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4508971);