Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470137 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2792 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)(7,11)(8,12)(9,13)(10,14)(15,18)(16,19)(17,20)(21,22,27,32)(23,33,35,45)(24,38,42,47)(25,31,26,30)(28,37,44,36)(29,41,48,40)(34,46,39,43)(49,51,52,50);(2,4)(5,7,10,17)(6,11,14,20)(8,16,9,15)(12,19,13,18)(21,23,27,35)(22,28,32,44)(24,39,42,34)(25,37,26,36)(29,46,48,43)(30,45,31,33)(38,49,47,52)(40,50,41,51);(5,8,10,9)(6,12,14,13)(7,15,17,16)(11,18,20,19)(21,24,27,42)(22,29,32,48)(23,34,35,39)(25,40,26,41)(28,43,44,46)(30,38,31,47)(33,49,45,52)(36,50,37,51);(5,9,10,8)(6,13,14,12)(7,16,17,15)(11,19,20,18)(23,35)(28,44)(33,45)(34,39)(36,37)(43,46)(49,52)(50,51);(6,14)(11,20)(12,13)(18,19)(21,25,27,26)(22,30,32,31)(23,36,35,37)(24,40,42,41)(28,33,44,45)(29,38,48,47)(34,50,39,51)(43,49,46,52)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6396892);