Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500443 |
| 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 | |
1688 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)(21,22,26,30)(23,31,34,47)(24,28,37,44)(25,39,32,29)(27,40,42,35)(33,46,36,50)(38,52,48,45)(41,51,43,49);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,32,40,26,34,25,35)(22,27,39,47,30,42,29,31)(24,36,48,49,37,33,38,51)(28,43,52,46,44,41,45,50);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24,26,37)(22,28,30,44)(23,33,34,36)(25,38,32,48)(27,41,42,43)(29,45,39,52)(31,46,47,50)(35,49,40,51);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(23,34)(27,42)(31,47)(33,36)(35,40)(41,43)(46,50)(49,51);(6,12)(10,13)(11,20)(18,19)(23,34)(24,37)(27,42)(28,44)(31,47)(35,40)(38,48)(45,52)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=h,d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6437062);