Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470355 |
| 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 | |
1776 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,26,29)(23,31,33,47)(24,28,36,44)(25,38,39,30)(27,40,42,34)(32,46,35,50)(37,52,48,45)(41,51,43,49);(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,25,34,26,33,39,40)(22,27,30,31,29,42,38,47)(24,35,37,51,36,32,48,49)(28,43,45,50,44,41,52,46);(2,4)(7,13)(10,17)(14,19)(18,20)(21,24)(22,28)(23,32)(25,37)(26,36)(27,41)(29,44)(30,45)(31,46)(33,35)(34,49)(38,52)(39,48)(40,51)(42,43)(47,50);(7,13)(10,17)(14,19)(18,20)(23,33)(24,36)(27,42)(28,44)(31,47)(34,40)(37,48)(45,52);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(22,29)(27,42)(28,44)(30,38)(31,47)(41,43)(45,52)(46,50)Polycyclic presentation:
2^9|a^2=h,b^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6403649);