Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470137 |
| 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×#3512 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2918 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)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22,27,32)(23,28,34,43)(24,36,39,48)(25,31,26,30)(29,41,46,38)(33,49,37,52)(35,47,40,44)(42,50,45,51);(2,4)(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23)(22,28)(24,37)(25,40)(26,35)(27,34)(29,45)(30,47)(31,44)(32,43)(33,39)(36,52)(38,50)(41,51)(42,46)(48,49);(7,13)(10,17)(14,15)(18,19)(21,24)(22,29)(23,33)(25,38)(26,41)(27,39)(28,42)(30,36)(31,48)(32,46)(34,37)(35,50)(40,51)(43,45)(44,52)(47,49);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(23,34)(28,43)(33,37)(35,40)(42,45)(44,47)(49,52)(50,51);(6,12)(10,17)(11,20)(18,19)(21,25,27,26)(22,30,32,31)(23,35,34,40)(24,38,39,41)(28,44,43,47)(29,36,46,48)(33,50,37,51)(42,52,45,49)Polycyclic presentation:
2^9|a^2=i,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,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, 6388605);