Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432843 |
| 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 | #29607 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2923 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)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20)(21,22)(23,31)(24,28)(25,38)(26,29)(27,33)(30,40)(32,47)(34,46)(35,49)(36,44)(37,52)(39,42)(41,48)(43,50)(45,51);(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(21,23,26,34)(22,27,29,42)(24,35,36,32)(25,39,40,33)(28,43,44,41)(30,46,38,31)(37,48,51,50)(45,47,52,49);(2,4)(7,13)(10,16)(14,19)(17,20)(21,24)(22,28)(23,32)(25,37)(26,36)(27,41)(29,44)(30,45)(31,47)(33,48)(34,35)(38,52)(39,50)(40,51)(42,43)(46,49);(6,11)(10,16)(12,18)(17,20)(22,29)(24,36)(27,42)(30,38)(31,46)(32,35)(37,51)(48,50);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)(24,36)(28,44)(32,35)(37,51)(41,43)(45,52)(47,49)(48,50)Polycyclic presentation:
2^9|b^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,f^b=fi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343767);