Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470161 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22,27,30)(23,28,33,44)(24,35,37,46)(25,31,26,39)(29,40,45,36)(32,47,48,52)(34,42,38,43)(41,49,51,50);(1,2,3,4)(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23)(22,28)(24,32)(25,38)(26,34)(27,33)(29,41)(30,44)(31,43)(35,47)(36,50)(37,48)(39,42)(40,49)(45,51)(46,52);(7,13)(10,17)(14,15)(18,19)(21,24)(22,29)(23,32)(25,36)(26,40)(27,37)(28,41)(30,45)(31,46)(33,48)(34,49)(35,39)(38,50)(42,47)(43,52)(44,51);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(22,30)(23,33)(29,45)(31,39)(32,48)(34,38)(35,46)(49,50);(6,12)(10,17)(11,20)(18,19)(21,25,27,26)(22,31,30,39)(23,34,33,38)(24,36,37,40)(28,42,44,43)(29,46,45,35)(32,49,48,50)(41,47,51,52)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6389006);