Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1765 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)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,25,29)(23,31,35,27)(24,36,37,28)(26,30,39,44)(32,42,34,40)(33,41,47,45)(38,49,50,43)(46,51,48,52);(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23,25,35)(22,27,29,31)(24,33,37,47)(26,32,39,34)(28,41,36,45)(30,40,44,42)(38,46,50,48)(43,51,49,52);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(23,32)(26,39)(27,40)(30,44)(31,42)(33,46)(34,35)(38,50)(41,51)(43,49)(45,52)(47,48);(6,12)(10,13)(11,18)(17,19)(21,24)(22,28)(23,33)(25,37)(26,38)(27,41)(29,36)(30,43)(31,45)(32,46)(34,48)(35,47)(39,50)(40,51)(42,52)(44,49);(7,15)(10,13)(14,20)(17,19)(23,34)(24,37)(26,39)(27,42)(28,36)(30,44)(31,40)(32,35)(33,46)(41,51)(45,52)(47,48)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6576932);