Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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 | C2×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1769 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)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22)(23,27)(24,34)(25,37)(26,29)(28,35)(30,39)(31,45)(32,41)(33,49)(36,44)(38,43)(40,47)(42,48)(46,52)(50,51);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,26,32)(22,27,29,41)(24,31,36,46)(25,33,39,48)(28,40,43,51)(30,42,37,49)(34,45,44,52)(35,47,38,50);(7,13)(10,16)(14,19)(17,20)(21,24,26,36)(22,28,29,43)(23,31,32,46)(25,38,39,35)(27,40,41,51)(30,44,37,34)(33,50,48,47)(42,52,49,45);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(22,29)(23,32)(28,43)(30,37)(31,46)(33,48)(34,44)(47,50);(8,15)(11,18)(14,19)(17,20)(23,32)(27,41)(31,46)(33,48)(40,51)(42,49)(45,52)(47,50)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=ei,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, 6471751);