Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1526 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)(21,22,26,27)(23,24,29,32)(25,28,34,35)(30,31,33,36)(37,38,42,44)(39,45,48,43)(40,49,51,41)(46,52,50,47);(6,10)(7,13)(12,17)(14,19)(22,27)(23,29)(28,35)(30,33)(37,39,42,48)(38,43,44,45)(40,50,51,46)(41,47,49,52);(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,23)(22,24)(25,33)(26,29)(27,32)(28,36)(30,34)(31,35)(37,39)(38,43)(40,46)(41,52)(42,48)(44,45)(47,49)(50,51);(1,2,3,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,24)(22,23)(25,31)(26,32)(27,29)(28,30)(33,35)(34,36)(37,40)(38,41)(39,46)(42,51)(43,52)(44,49)(45,47)(48,50);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,25,26,34)(22,28,27,35)(23,30,29,33)(24,31,32,36)(37,41)(38,40)(39,47)(42,49)(43,50)(44,51)(45,46)(48,52)Polycyclic presentation:
2^9|a^2=f,b^2=fgh,c^2=fh,d^2=hi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4575603);