Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 | #2615 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1810 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)(7,15)(8,12)(9,14)(10,13)(11,17)(16,20)(18,19)(21,22,25,27)(23,31,32,44)(24,35,36,29)(26,39,38,30)(28,40,41,34)(33,47,46,45)(37,43,51,50)(42,48,49,52);(1,2)(3,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,15)(14,20)(22,27)(23,32)(24,36)(30,39)(34,40)(37,51)(42,49)(45,47);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(21,23,25,32)(22,28,27,41)(24,33,36,46)(26,40,38,34)(29,42,35,49)(30,44,39,31)(37,52,51,48)(43,47,50,45);(6,13)(11,15)(12,14)(19,20)(21,24,25,36)(22,29,27,35)(23,33,32,46)(26,37,38,51)(28,42,41,49)(30,43,39,50)(31,45,44,47)(34,48,40,52);(2,4)(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(24,36)(29,35)(33,46)(37,51)(42,49)(43,50)(45,47)(48,52)Polycyclic presentation:
2^9|a^2=i,c^2=f,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6581707);