Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1636 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)(5,6,9,13)(7,11,14,10)(8,16,18,12)(15,19,17,20)(21,22)(23,28)(24,34)(25,30)(26,41)(27,31)(29,35)(32,45)(33,42)(36,50)(37,43)(38,51)(39,47)(40,52)(44,49)(46,48);(1,2)(3,4)(5,7,9,14)(6,10,13,11)(8,17,18,15)(12,20,16,19)(21,23,27,26)(22,28,31,41)(24,35,37,32)(25,33,39,40)(29,43,45,34)(30,42,47,52)(36,49,38,48)(44,51,46,50);(5,7)(6,11)(8,15)(9,14)(10,13)(12,20)(16,19)(17,18)(21,24,23,32,27,37,26,35)(22,29,41,43,31,45,28,34)(25,38,33,49,39,36,40,48)(30,46,52,50,47,44,42,51);(2,4)(7,14)(10,11)(15,17)(19,20)(21,25)(22,30)(23,33)(24,36)(26,40)(27,39)(28,42)(29,44)(31,47)(32,48)(34,50)(35,49)(37,38)(41,52)(43,51)(45,46);(5,8,9,18)(6,12,13,16)(7,15,14,17)(10,19,11,20)(22,31)(25,39)(28,41)(29,45)(33,40)(34,43)(36,38)(48,49)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7100244);