Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432861 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2844 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)(21,22,26,30)(23,31,32,44)(24,35,36,28)(25,38,40,29)(27,39,41,34)(33,47,46,45)(37,43,51,50)(42,48,49,52);(5,7,9,14)(6,10,12,18)(8,15,16,13)(11,19,20,17)(21,23,26,32)(22,27,30,41)(24,33,36,46)(25,39,40,34)(28,42,35,49)(29,44,38,31)(37,52,51,48)(43,47,50,45);(2,4)(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(23,32)(27,41)(31,44)(33,46)(34,39)(42,49)(45,47)(48,52);(6,12)(10,18)(11,20)(17,19)(21,24)(22,28)(23,33)(25,37)(26,36)(27,42)(29,43)(30,35)(31,45)(32,46)(34,48)(38,50)(39,52)(40,51)(41,49)(44,47);(24,36)(28,35)(33,46)(37,51)(42,49)(43,50)(45,47)(48,52)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,f^b=fi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341890);