Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470374 |
| 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 | #2615 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1785 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)(21,22,27,32)(23,33,35,47)(24,29,39,46)(25,31,26,30)(28,40,43,36)(34,49,37,52)(38,48,41,45)(42,50,44,51);(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23,26,36,27,35,25,40)(22,28,31,33,32,43,30,47)(24,37,38,50,39,34,41,51)(29,44,45,52,46,42,48,49);(2,4)(7,13)(10,17)(14,15)(18,19)(21,24)(22,29)(23,34)(25,41)(26,38)(27,39)(28,42)(30,48)(31,45)(32,46)(33,49)(35,37)(36,51)(40,50)(43,44)(47,52);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(23,35)(28,43)(33,47)(34,37)(36,40)(42,44)(49,52)(50,51);(6,12)(10,17)(11,20)(18,19)(21,25,27,26)(22,30,32,31)(23,36,35,40)(24,38,39,41)(28,33,43,47)(29,45,46,48)(34,50,37,51)(42,52,44,49)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=ei,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6389072);