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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1702 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)(21,22,26,31,27,32,25,28)(23,29,35,42,33,44,39,40)(24,30,36,45,38,46,37,41)(34,43,49,51,47,52,48,50);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(22,28)(23,33)(24,36)(25,26)(29,42)(31,32)(34,48)(37,38)(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)(22,29)(24,34)(25,39)(26,35)(27,33)(28,40)(30,43)(31,42)(32,44)(36,49)(37,48)(38,47)(41,50)(45,51)(46,52);(7,14)(10,13)(15,20)(18,19)(21,24,25,37,27,38,26,36)(22,30,28,41,32,46,31,45)(23,34,39,48,33,47,35,49)(29,43,40,50,44,52,42,51);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,25,27,26)(22,31,32,28)(23,35,33,39)(24,37,38,36)(29,40,44,42)(30,45,46,41)(34,49,47,48)(43,50,52,51)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=g,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, 7093889);