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 | |
1570 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)(8,15)(18,20)(21,22,25,26)(23,29,30,27)(24,32,33,28)(31,34,36,35)(37,38)(39,44)(40,47)(41,43)(42,45)(46,51)(48,52)(49,50);(6,10)(7,13)(12,17)(14,19)(22,26)(23,30)(24,33)(34,35)(37,39,43,40)(38,44,41,47)(42,46,50,48)(45,51,49,52);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,22)(23,29)(24,28)(25,26)(27,30)(31,35)(32,33)(34,36)(37,40,43,39)(38,44,41,47)(42,48,50,46)(45,51,49,52);(1,2,3,4)(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,23,25,30)(22,27,26,29)(24,31,33,36)(28,34,32,35)(37,41,43,38)(39,44,40,47)(42,49,50,45)(46,51,48,52);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,24,25,33)(22,28,26,32)(23,31,30,36)(27,34,29,35)(37,42,43,50)(38,45,41,49)(39,46,40,48)(44,51,47,52)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=gh,d^2=i,e^2=g,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, 4408029);