Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470149 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2958 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22,27,32)(23,33,35,44)(24,38,41,45)(25,30,26,31)(28,37,43,36)(29,40,46,39)(34,48,49,42)(47,51,52,50);(2,4)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,35)(22,28,32,43)(24,34,41,49)(25,37,26,36)(29,42,46,48)(30,33,31,44)(38,52,45,47)(39,51,40,50);(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24)(22,29)(23,34)(25,39)(26,40)(27,41)(28,42)(30,45)(31,38)(32,46)(33,47)(35,49)(36,50)(37,51)(43,48)(44,52);(7,14)(10,18)(13,15)(17,19)(23,35)(28,43)(33,44)(34,49)(36,37)(42,48)(47,52)(50,51);(6,12)(10,18)(11,20)(17,19)(21,25,27,26)(22,30,32,31)(23,36,35,37)(24,39,41,40)(28,44,43,33)(29,45,46,38)(34,50,49,51)(42,52,48,47)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,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=dhi,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6384717);