Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | |
1768 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)(23,32)(24,29)(25,30)(26,40)(27,31)(28,35)(33,46)(34,42)(36,47)(37,43)(38,45)(39,44)(41,48)(49,51)(50,52);(2,3)(7,13)(8,16)(10,17)(11,15);(5,7,9,13)(6,10,12,17)(8,14,16,19)(11,18,15,20)(21,23,25,34,27,36,26,35)(22,28,40,47,31,42,30,32)(24,33,39,50,38,49,37,48)(29,41,43,51,45,52,44,46);(1,3,4,2)(7,13)(10,17)(14,19)(18,20)(21,24,27,38)(22,29,31,45)(23,33,36,49)(25,39,26,37)(28,41,42,52)(30,44,40,43)(32,46,47,51)(34,50,35,48);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,25,27,26)(22,30,31,40)(23,34,36,35)(24,37,38,39)(28,32,42,47)(29,43,45,44)(33,48,49,50)(41,51,52,46)Polycyclic presentation:
2^9|a^2=fh,c^2=ghi,d^2=fi,e^2=hi,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=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7103896);