Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1572 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)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,22,25,28)(23,29,30,26)(24,32,33,27)(31,34,36,35)(37,38,42,46)(39,40,44,43)(41,45,50,52)(47,48,49,51);(6,10)(8,16)(11,13)(14,20)(23,30)(24,33)(26,29)(27,32)(37,39,42,44)(38,43,46,40)(41,47,50,49)(45,51,52,48);(5,6)(7,11)(8,17)(9,10)(12,16)(13,15)(14,19)(18,20)(21,23,25,30)(22,26,28,29)(24,31,33,36)(27,34,32,35)(37,39)(38,43)(40,46)(41,49)(42,44)(45,48)(47,50)(51,52);(1,2)(3,4)(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19)(21,22)(23,29)(24,27)(25,28)(26,30)(31,35)(32,33)(34,36)(37,40,42,43)(38,44,46,39)(41,48,50,51)(45,49,52,47);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,24,25,33)(22,27,28,32)(23,31,30,36)(26,34,29,35)(37,41)(38,45)(39,47)(40,48)(42,50)(43,51)(44,49)(46,52)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=fh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5382969);