Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500416 |
| 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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2005 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)(23,33)(24,38)(25,30)(26,31)(27,32)(28,37)(29,42)(34,43)(35,45)(36,44)(39,46)(40,48)(41,47)(49,50)(51,52);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,25,37,27,35,26,36)(22,28,31,33,32,44,30,45)(24,39,41,50,40,34,42,51)(29,46,48,52,47,43,38,49);(7,13)(8,15)(10,16)(11,18)(21,24,26,42,27,40,25,41)(22,29,30,38,32,47,31,48)(23,34,36,50,35,39,37,51)(28,43,45,52,44,46,33,49);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(23,35)(24,40)(28,44)(29,47)(33,45)(36,37)(38,48)(41,42);(7,13)(10,16)(14,19)(17,20)(21,25,27,26)(22,30,32,31)(23,36,35,37)(24,41,40,42)(28,33,44,45)(29,38,47,48)(34,50,39,51)(43,49,46,52)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=g,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6434146);