Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22)(23,34)(24,30)(25,41)(26,32)(27,33)(28,31)(29,38)(35,52)(36,45)(37,49)(39,51)(40,50)(42,47)(43,48)(44,46);(5,7,9,15)(6,10,12,13)(8,14,17,20)(11,18,16,19)(21,23)(22,29)(24,39)(25,36)(26,43)(27,38)(28,44)(30,47)(31,45)(32,50)(33,34)(35,40)(37,42)(41,46)(48,52)(49,51);(2,4)(21,24,27,37)(22,30,33,49)(23,35,38,26)(25,40,28,43)(29,32,34,52)(31,48,41,50)(36,42,44,39)(45,47,46,51);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,25,27,28)(22,31,33,41)(23,36,38,44)(24,40,37,43)(26,39,35,42)(29,45,34,46)(30,48,49,50)(32,47,52,51);(7,15)(10,13)(14,20)(18,19)(21,26,27,35)(22,32,33,52)(23,37,38,24)(25,42,28,39)(29,30,34,49)(31,51,41,47)(36,40,44,43)(45,50,46,48)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6579565);