Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1540 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,26,25)(23,29,30,27)(24,32,33,28)(31,34,36,35)(37,38,41,44)(39,45,47,42)(40,48,49,43)(46,50,52,51);(7,13)(10,17)(14,19)(18,20)(22,25)(23,30)(24,33)(34,35)(37,38,41,44)(39,42,47,45)(40,48,49,43)(46,51,52,50);(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,26,30)(22,27,25,29)(24,31,33,36)(28,34,32,35)(37,38)(39,45)(40,43)(41,44)(42,47)(46,51)(48,49)(50,52);(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,16)(19,20)(21,24,26,33)(22,28,25,32)(23,31,30,36)(27,34,29,35)(37,39,41,47)(38,42,44,45)(40,46,49,52)(43,50,48,51);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,25,26,22)(23,27,30,29)(24,32,33,28)(31,34,36,35)(37,40,41,49)(38,43,44,48)(39,46,47,52)(42,50,45,51)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=f,d^2=ghi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5191841);