Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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 | |
1801 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)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22)(23,31)(24,29)(25,27)(26,38)(28,34)(30,37)(32,45)(33,44)(35,41)(36,49)(39,40)(42,47)(43,48)(46,52)(50,51);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(22,27)(28,40)(29,41)(30,38)(31,44)(42,51)(43,49)(45,52);(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,23,25,33)(22,28,27,40)(24,32,35,46)(26,39,37,34)(29,42,41,51)(30,44,38,31)(36,50,48,47)(43,52,49,45);(8,16)(11,20)(13,15)(17,19)(21,24,25,35)(22,29,27,41)(23,32,33,46)(26,36,37,48)(28,42,40,51)(30,43,38,49)(31,45,44,52)(34,47,39,50);(7,14)(10,18)(13,15)(17,19)(24,35)(29,41)(32,46)(36,48)(42,51)(43,49)(45,52)(47,50)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6416142);