Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
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)(6,10)(7,13)(12,17)(14,19)(21,22,26,27)(23,24,31,28)(25,29,33,34)(30,32,36,35)(37,38,41,45)(39,43,47,42)(40,48,50,44)(46,51,49,52);(7,13)(11,16)(14,19)(18,20)(22,27)(24,28)(25,33)(30,36)(37,39,41,47)(38,42,45,43)(40,49,50,46)(44,52,48,51);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,23,26,31)(22,28,27,24)(25,30,33,36)(29,35,34,32)(37,39)(38,43)(40,46)(41,47)(42,45)(44,52)(48,51)(49,50);(1,2,3,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,24,26,28)(22,23,27,31)(25,32,33,35)(29,30,34,36)(37,38)(39,42)(40,44)(41,45)(43,47)(46,51)(48,50)(49,52);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,25)(22,29)(23,30)(24,32)(26,33)(27,34)(28,35)(31,36)(37,40,41,50)(38,44,45,48)(39,46,47,49)(42,51,43,52)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fh,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4408089);