Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432843 |
| 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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1718 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)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,26,29)(23,27,32,42)(24,34,37,46)(25,38,40,30)(28,39,45,36)(31,47,35,52)(33,50,49,43)(41,51,44,48);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23)(22,27)(24,35)(25,33)(26,32)(28,44)(29,42)(30,43)(31,37)(34,52)(36,51)(38,50)(39,48)(40,49)(41,45)(46,47);(7,13)(10,17)(14,19)(18,20)(21,24,26,37)(22,28,29,45)(23,31,32,35)(25,39,40,36)(27,41,42,44)(30,46,38,34)(33,51,49,48)(43,52,50,47);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(22,29)(27,42)(28,45)(30,38)(34,46)(41,44)(43,50)(47,52);(23,32)(27,42)(31,35)(33,49)(41,44)(43,50)(47,52)(48,51)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,d^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,e^b=ei,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343359);