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×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
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)(23,33,35,45)(24,38,42,47)(25,31,26,30)(28,37,44,36)(29,41,48,40)(34,46,39,43)(49,51,52,50);(2,4)(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23)(22,28)(24,39)(25,37)(26,36)(27,35)(29,46)(30,45)(31,33)(32,44)(34,42)(38,49)(40,50)(41,51)(43,48)(47,52);(7,13)(10,17)(14,15)(18,19)(21,24)(22,29)(23,34)(25,40)(26,41)(27,42)(28,43)(30,38)(31,47)(32,48)(33,49)(35,39)(36,50)(37,51)(44,46)(45,52);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(23,35)(28,44)(33,45)(34,39)(36,37)(43,46)(49,52)(50,51);(6,12)(10,17)(11,20)(18,19)(21,25,27,26)(22,30,32,31)(23,36,35,37)(24,40,42,41)(28,33,44,45)(29,38,48,47)(34,50,39,51)(43,49,46,52)Polycyclic presentation:
2^9|a^2=i,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6388885);