Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | #2397 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,4,3)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22)(23,31)(24,29)(25,27)(26,39)(28,35)(30,32)(33,46)(34,45)(36,41)(37,50)(38,52)(40,47)(42,49)(43,51)(44,48);(2,3)(7,13)(8,16)(10,17)(11,15)(22,27)(24,36)(28,40)(30,39)(31,45)(33,37)(38,48)(49,51);(5,7,9,13)(6,10,12,17)(8,14,16,19)(11,18,15,20)(21,23,32,47,25,34,26,35)(22,28,39,45,27,40,30,31)(24,37,48,49,36,33,38,51)(29,43,52,46,41,42,44,50);(1,3,4,2)(7,13)(10,17)(14,19)(18,20)(21,24)(22,29)(23,33)(25,36)(26,38)(27,41)(28,42)(30,44)(31,46)(32,48)(34,37)(35,49)(39,52)(40,43)(45,50)(47,51);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(24,36)(29,41)(33,37)(38,48)(42,43)(44,52)(46,50)(49,51)Polycyclic presentation:
2^9|a^2=fhi,c^2=ghi,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7103903);