Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
2730 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)(23,34)(24,38)(25,31)(26,32)(27,33)(28,43)(29,37)(30,40)(35,50)(36,47)(39,45)(41,49)(42,46)(44,51)(48,52);(2,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23)(22,29)(24,39)(25,35)(26,42)(27,37)(28,36)(30,48)(31,45)(32,47)(33,34)(38,50)(40,51)(41,52)(43,46)(44,49);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24)(22,30)(23,35)(25,27)(26,40)(28,41)(29,44)(31,32)(33,49)(34,48)(36,51)(37,39)(38,43)(42,52)(45,46)(47,50);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,25,27,24)(22,31,33,38)(23,35,37,39)(26,40,28,41)(29,45,34,50)(30,43,49,32)(36,52,42,51)(44,47,48,46);(6,12)(10,13)(11,20)(18,19)(21,26,27,28)(22,32,33,43)(23,36,37,42)(24,40,25,41)(29,46,34,47)(30,31,49,38)(35,51,39,52)(44,45,48,50)Polycyclic presentation:
2^9|a^2=fi,c^2=hi,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436851);