Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470161 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2876 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)(21,22,27,30)(23,33,35,45)(24,38,42,48)(25,32,26,31)(28,37,44,36)(29,40,47,41)(34,46,39,43)(49,50,52,51);(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)(30,44)(31,45)(32,33)(34,42)(38,49)(40,50)(41,51)(43,47)(48,52);(7,13)(10,17)(14,15)(18,19)(21,24,27,42)(22,29,30,47)(23,34,35,39)(25,41,26,40)(28,43,44,46)(31,38,32,48)(33,49,45,52)(36,51,37,50);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(22,30)(28,44)(29,47)(31,32)(33,45)(38,48)(43,46)(49,52);(6,12)(10,17)(11,20)(18,19)(22,30)(23,35)(29,47)(31,32)(34,39)(36,37)(38,48)(50,51)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6388913);