Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535815 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,3,4)(21,22)(23,32)(24,29)(25,30)(26,40)(27,31)(28,35)(33,47)(34,43)(36,42)(37,44)(38,45)(39,46)(41,48)(49,51)(50,52);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,25,34,27,36,26,35)(22,28,40,42,31,43,30,32)(24,33,38,50,39,49,37,48)(29,41,44,51,46,52,45,47);(2,4)(6,10)(11,17)(12,18)(19,20)(21,24,27,39)(22,29,31,46)(23,33,36,49)(25,38,26,37)(28,41,43,52)(30,45,40,44)(32,47,42,51)(34,50,35,48);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,25,27,26)(22,30,31,40)(23,34,36,35)(24,37,39,38)(28,32,43,42)(29,44,46,45)(33,48,49,50)(41,51,52,47);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,25,27,26)(22,30,31,40)(23,35,36,34)(24,38,39,37)(28,42,43,32)(29,45,46,44)(33,48,49,50)(41,51,52,47)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,d^2=hi,e^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ei,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6454380);