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 | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1821 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)(24,35)(25,27)(26,39)(28,34)(29,36)(30,38)(32,46)(33,44)(37,43)(40,41)(42,52)(45,47)(48,49)(50,51);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(22,27)(28,41)(29,35)(30,39)(31,44)(42,49)(43,50)(45,46);(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,23,25,33)(22,28,27,41)(24,32,36,47)(26,40,38,34)(29,42,35,49)(30,44,39,31)(37,52,51,48)(43,46,50,45);(8,16)(11,20)(13,15)(17,19)(21,24,25,36)(22,29,27,35)(23,32,33,47)(26,37,38,51)(28,42,41,49)(30,43,39,50)(31,45,44,46)(34,48,40,52);(7,14)(10,18)(13,15)(17,19)(24,36)(29,35)(32,47)(37,51)(42,49)(43,50)(45,46)(48,52)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,d^a=di,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, 6416158);