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 | #42679 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1798 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)(7,15)(8,12)(9,13)(10,14)(11,17)(16,20)(18,19)(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)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23)(22,27)(24,33)(25,35)(26,32)(28,41)(29,31)(30,40)(34,39)(36,45)(37,47)(38,46)(42,44)(43,51)(48,50)(49,52);(1,2,3,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(23,32)(26,39)(27,40)(30,44)(31,42)(33,46)(34,35)(38,50)(41,51)(43,49)(45,52)(47,48);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(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);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(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,c^2=fgi,d^2=fi,e^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, 6594625);