Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1695 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,26,30)(23,31,34,47)(24,28,36,43)(25,39,32,29)(27,40,42,35)(33,46,37,50)(38,52,48,45)(41,51,44,49);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(24,36)(28,43)(33,37)(38,48)(41,44)(45,52)(46,50)(49,51);(2,4)(21,23,32,40,26,34,25,35)(22,27,39,47,30,42,29,31)(24,37,48,49,36,33,38,51)(28,44,52,46,43,41,45,50);(6,11)(10,13)(12,19)(18,20)(21,24,26,36)(22,28,30,43)(23,33,34,37)(25,38,32,48)(27,41,42,44)(29,45,39,52)(31,46,47,50)(35,49,40,51);(2,4)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(23,34)(24,36)(27,42)(28,43)(31,47)(35,40)(38,48)(45,52)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593277);