Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | |
1638 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)(21,22,26,30)(23,31,34,45)(24,28,36,42)(25,38,32,29)(27,39,41,35)(33,44,48,52)(37,49,46,43)(40,50,51,47);(1,2)(3,4)(5,7,9,14)(6,10,13,11)(8,17,18,15)(12,20,16,19);(5,7)(6,11)(8,15)(9,14)(10,13)(12,20)(16,19)(17,18)(21,23,32,39,26,34,25,35)(22,27,38,45,30,41,29,31)(24,33,46,50,36,48,37,47)(28,40,49,52,42,51,43,44);(2,4)(7,14)(10,11)(15,17)(19,20)(21,24)(22,28)(23,33)(25,37)(26,36)(27,40)(29,43)(30,42)(31,44)(32,46)(34,48)(35,47)(38,49)(39,50)(41,51)(45,52);(5,8,9,18)(6,12,13,16)(7,15,14,17)(10,19,11,20)(23,34)(24,36)(27,41)(28,42)(31,45)(35,39)(37,46)(43,49)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 7100467);