Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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 | C2×#3565 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2765 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)(21,22)(23,33)(24,38)(25,30)(26,31)(27,32)(28,36)(29,39)(34,42)(35,43)(37,46)(40,44)(41,45)(47,48)(49,51)(50,52);(2,4)(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,27,37)(22,28,32,43)(24,34,40,50)(25,36,26,35)(29,42,45,52)(30,46,31,33)(38,47,44,51)(39,49,41,48);(6,10)(11,17)(12,18)(19,20)(21,24)(22,29)(23,34)(25,39)(26,41)(27,40)(28,42)(30,44)(31,38)(32,45)(33,47)(35,48)(36,49)(37,50)(43,52)(46,51);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,25,27,26)(22,30,32,31)(23,35,37,36)(24,39,40,41)(28,33,43,46)(29,44,45,38)(34,48,50,49)(42,47,52,51);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,25,27,26)(22,31,32,30)(23,36,37,35)(24,39,40,41)(28,33,43,46)(29,38,45,44)(34,49,50,48)(42,47,52,51)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,d^2=hi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6451432);