Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6)(7,14)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22,26,30)(23,27,32,41)(24,34,37,51)(25,39,36,29)(28,43,45,38)(31,46,35,50)(33,49,47,42)(40,52,44,48);(2,3)(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,23)(22,27)(24,35)(25,33)(26,32)(28,44)(29,42)(30,41)(31,37)(34,50)(36,47)(38,52)(39,49)(40,45)(43,48)(46,51);(1,2,4,3)(21,24,36,43,26,37,25,38)(22,28,39,51,30,45,29,34)(23,31,47,52,32,35,33,48)(27,40,49,50,41,44,42,46);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(23,32)(24,37)(27,41)(28,45)(33,47)(34,51)(38,43)(42,49);(1,3,4,2)(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(23,32)(27,41)(31,35)(33,47)(40,44)(42,49)(46,50)(48,52)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6723845);