Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | #29669 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1736 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)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22,26,31)(23,28,34,43)(24,36,39,29)(25,30,35,44)(27,32,38,46)(33,49,51,42)(37,47,40,45)(41,52,50,48);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,26,34)(22,28,31,43)(24,37,39,40)(25,38,35,27)(29,45,36,47)(30,46,44,32)(33,50,51,41)(42,52,49,48);(2,4)(21,24,38,50,26,39,27,41)(22,29,46,52,31,36,32,48)(23,33,25,40,34,51,35,37)(28,42,30,47,43,49,44,45);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(24,39)(29,36)(33,51)(37,40)(41,50)(42,49)(45,47)(48,52);(7,15)(10,13)(14,20)(18,19)(21,25,27,23,26,35,38,34)(22,30,32,28,31,44,46,43)(24,40,41,33,39,37,50,51)(29,47,48,42,36,45,52,49)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=hi,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715777);