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)(7,11)(8,10)(9,16)(12,18)(13,20)(14,15)(17,19)(21,22,26,25)(23,29,30,27)(24,32,33,28)(31,34,36,35)(37,38)(39,45)(40,43)(41,44)(42,47)(46,51)(48,49)(50,52);(6,8)(11,15)(12,16)(19,20)(22,25)(23,30)(24,33)(34,35)(37,38,41,44)(39,45,47,42)(40,43,49,48)(46,51,52,50);(5,7,10,14)(6,11,8,15)(9,17,18,13)(12,20,16,19)(21,23,26,30)(22,27,25,29)(24,31,33,36)(28,34,32,35)(37,38)(39,45)(40,48)(41,44)(42,47)(43,49)(46,50)(51,52);(1,2,3,4)(5,8,10,6)(7,11,14,15)(9,16,18,12)(13,19,17,20)(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);(5,9,10,18)(6,12,8,16)(7,13,14,17)(11,19,15,20)(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=g,b^2=gh,c^2=fgh,d^2=i,e^2=f,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, 5187663);