Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500456 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1966 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)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22)(23,31)(24,29)(25,37)(26,28)(27,34)(30,38)(32,44)(33,43)(35,41)(36,49)(39,47)(40,46)(42,48)(45,51)(50,52);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,25,34,26,32,38,46)(22,27,30,31,28,40,37,44)(24,33,36,47,35,45,48,52)(29,39,42,43,41,50,49,51);(1,2,3,4)(7,13)(8,15)(10,16)(11,18)(22,28)(23,32)(29,41)(30,37)(33,45)(34,46)(42,49)(47,52);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(21,24)(22,29)(23,33)(25,36)(26,35)(27,39)(28,41)(30,42)(31,43)(32,45)(34,47)(37,49)(38,48)(40,50)(44,51)(46,52);(7,13)(10,16)(14,19)(17,20)(24,35)(29,41)(33,45)(36,48)(39,50)(42,49)(43,51)(47,52)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=gi,d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6434112);