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×#3493 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2972 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)(23,33,35,44)(24,38,39,46)(25,31,26,30)(28,37,43,36)(29,40,45,41)(34,48,49,42)(47,50,52,51);(2,4)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23)(22,28)(24,34)(25,37)(26,36)(27,35)(29,42)(30,44)(31,33)(32,43)(38,52)(39,49)(40,51)(41,50)(45,48)(46,47);(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24,27,39)(22,29,32,45)(23,34,35,49)(25,41,26,40)(28,42,43,48)(30,38,31,46)(33,47,44,52)(36,51,37,50);(7,14)(10,18)(13,15)(17,19)(23,35)(24,39)(28,43)(29,45)(33,44)(36,37)(38,46)(40,41);(6,12)(10,18)(11,20)(17,19)(23,35)(28,43)(33,44)(34,49)(36,37)(42,48)(47,52)(50,51)Polycyclic presentation:
2^9|a^2=i,c^2=hi,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=dhi,e^b=ei,f^b=fi,g^b=gi,d^c=di,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, 6384735);