Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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)(5,6,10,8)(7,11,13,14)(9,16,17,12)(15,20,19,18)(21,22,26,27)(23,28,29,24)(25,30,34,35)(31,33,36,32)(37,38,41,44)(39,45,47,42)(40,48,49,43)(46,50,52,51);(7,13)(11,14)(15,19)(18,20)(22,27)(23,29)(30,35)(31,36)(37,38,41,44)(39,42,47,45)(40,48,49,43)(46,51,52,50);(5,7,10,13)(6,11,8,14)(9,15,17,19)(12,18,16,20)(21,23)(22,28)(24,27)(25,31)(26,29)(30,33)(32,35)(34,36)(37,38)(39,45)(40,43)(41,44)(42,47)(46,51)(48,49)(50,52);(1,2)(3,4)(5,8,10,6)(7,14,13,11)(9,16,17,12)(15,20,19,18)(21,24,26,28)(22,29,27,23)(25,33,34,32)(30,31,35,36)(37,39)(38,42)(40,46)(41,47)(43,50)(44,45)(48,51)(49,52);(5,9)(6,12)(7,15)(8,16)(10,17)(11,18)(13,19)(14,20)(21,25)(22,30)(23,31)(24,32)(26,34)(27,35)(28,33)(29,36)(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=g,c^2=fgh,d^2=gh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5527207);