Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500416 |
| 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 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1734 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)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22,26,30)(23,27,32,40)(24,34,35,49)(25,37,38,29)(28,42,43,36)(31,44,45,52)(33,48,47,41)(39,50,51,46);(2,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23)(22,27)(24,31)(25,33)(26,32)(28,39)(29,41)(30,40)(34,44)(35,45)(36,46)(37,48)(38,47)(42,50)(43,51)(49,52);(7,13)(8,15)(10,16)(11,18)(21,24,25,36,26,35,38,42)(22,28,29,34,30,43,37,49)(23,31,33,46,32,45,47,50)(27,39,41,44,40,51,48,52);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(23,32)(27,40)(31,45)(33,47)(39,51)(41,48)(44,52)(46,50);(7,13)(10,16)(14,19)(17,20)(24,35)(28,43)(31,45)(34,49)(36,42)(39,51)(44,52)(46,50)Polycyclic presentation:
2^9|a^2=hi,c^2=g,d^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6433820);