Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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×#469 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ 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:
(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,26,28)(23,32,33,44)(24,36,38,30)(25,31,27,41)(29,39,42,35)(34,49,48,47)(37,46,40,45)(43,51,52,50);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(22,28)(23,33)(30,36)(31,41)(34,48)(35,39)(45,46)(50,51);(2,4)(21,23,25,39,26,33,27,35)(22,29,41,44,28,42,31,32)(24,34,40,50,38,48,37,51)(30,43,45,49,36,52,46,47);(6,11)(10,13)(12,19)(18,20)(21,24,26,38)(22,30,28,36)(23,34,33,48)(25,40,27,37)(29,43,42,52)(31,46,41,45)(32,47,44,49)(35,51,39,50);(2,4)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(21,25,26,27)(22,31,28,41)(23,35,33,39)(24,37,38,40)(29,44,42,32)(30,45,36,46)(34,50,48,51)(43,47,52,49)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593231);