Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1745 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,26,31)(23,27,35,44)(24,29,36,47)(25,38,40,30)(28,39,45,37)(32,46,33,42)(34,50,51,43)(41,49,52,48);(1,2)(3,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23)(22,27)(24,32)(25,34)(26,35)(28,41)(29,46)(30,43)(31,44)(33,36)(37,48)(38,50)(39,49)(40,51)(42,47)(45,52);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,24)(22,28)(23,32)(25,39)(26,36)(27,41)(29,38)(30,47)(31,45)(33,35)(34,49)(37,40)(42,43)(44,52)(46,50)(48,51);(6,12)(10,13)(11,18)(17,19)(21,24)(22,29)(23,33)(25,37)(26,36)(27,42)(28,38)(30,45)(31,47)(32,35)(34,49)(39,40)(41,43)(44,46)(48,51)(50,52);(2,4)(7,15)(10,13)(14,20)(17,19)(24,36)(28,45)(29,47)(32,33)(37,39)(41,52)(42,46)(48,49)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6577232);