Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| 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×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1777 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)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,27,31)(23,28,34,44)(24,35,38,47)(25,30,26,40)(29,41,48,37)(32,49,36,52)(33,43,39,45)(42,50,46,51);(2,3)(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23,27,34)(22,28,31,44)(24,36,38,32)(25,39,26,33)(29,46,48,42)(30,45,40,43)(35,52,47,49)(37,50,41,51);(1,2,4,3)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,24,27,38)(22,29,31,48)(23,32,34,36)(25,37,26,41)(28,42,44,46)(30,47,40,35)(33,50,39,51)(43,49,45,52);(6,12)(10,13)(11,18)(17,19)(21,25,27,26)(22,30,31,40)(23,33,34,39)(24,37,38,41)(28,43,44,45)(29,47,48,35)(32,50,36,51)(42,49,46,52);(1,3,4,2)(7,15)(10,13)(14,20)(17,19)(23,34)(24,38)(28,44)(29,48)(33,39)(35,47)(37,41)(43,45)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fh,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6714982);