Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,4,3)(7,13)(11,16)(14,19)(18,20)(21,22,25,28)(23,26,29,33)(24,31,32,27)(30,36,35,34)(37,38)(39,45)(40,42)(41,48)(43,46)(44,49)(47,50)(51,52);(1,3,4,2)(6,10)(11,16)(12,17)(18,20)(23,29)(26,33)(30,35)(34,36)(38,40)(39,46)(41,49)(50,51);(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,32)(35,36)(37,39)(38,43)(40,45)(41,47)(42,46)(44,50)(48,51)(49,52);(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,23,25,29)(22,26,28,33)(24,30,32,35)(27,34,31,36)(37,40,42,38)(39,43,46,45)(41,48,49,44)(47,50,52,51);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,24,25,32)(22,27,28,31)(23,30,29,35)(26,34,33,36)(37,41)(38,44)(39,47)(40,48)(42,49)(43,50)(45,51)(46,52)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,c^2=fh,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 496602);