Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3341 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(8,16)(11,17)(15,20)(18,19)(21,22,27,28,26,32,25,31)(23,29,35,40,33,44,36,42)(24,30,39,41,38,46,37,45)(34,43,48,50,47,52,49,51);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(22,28)(23,33)(24,37)(25,27)(29,42)(31,32)(34,48)(38,39)(40,44)(41,45)(43,52)(47,49);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,23,26,33)(22,29,32,44)(24,34,38,47)(25,36,27,35)(28,40,31,42)(30,43,46,52)(37,49,39,48)(41,50,45,51);(7,14)(10,13)(15,20)(18,19)(21,24,27,39,26,38,25,37)(22,30,28,41,32,46,31,45)(23,34,35,48,33,47,36,49)(29,43,40,50,44,52,42,51);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,25,26,27)(22,28,32,31)(23,35,33,36)(24,37,38,39)(29,42,44,40)(30,41,46,45)(34,48,47,49)(43,51,52,50)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,c^2=gi,d^2=fi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7093894);