Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1757 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)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,27,31)(23,28,34,44)(24,35,37,45)(25,30,26,39)(29,40,46,36)(32,47,49,52)(33,42,38,43)(41,48,51,50);(2,3)(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23)(22,28)(24,32)(25,38)(26,33)(27,34)(29,41)(30,43)(31,44)(35,47)(36,50)(37,49)(39,42)(40,48)(45,52)(46,51);(1,2,4,3)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,24)(22,29)(23,32)(25,36)(26,40)(27,37)(28,41)(30,45)(31,46)(33,48)(34,49)(35,39)(38,50)(42,47)(43,52)(44,51);(6,12)(10,13)(11,18)(17,19)(21,25,27,26)(22,30,31,39)(23,33,34,38)(24,36,37,40)(28,42,44,43)(29,45,46,35)(32,48,49,50)(41,47,51,52);(1,3,4,2)(7,15)(10,13)(14,20)(17,19)(24,37)(29,46)(32,49)(35,45)(36,40)(41,51)(47,52)(48,50)Polycyclic presentation:
2^9|a^2=i,c^2=fh,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6714973);