Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499792 |
| 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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22)(23,28)(24,36)(25,30)(26,39)(27,31)(29,37)(32,47)(33,42)(34,43)(35,44)(38,45)(40,46)(41,49)(48,52)(50,51);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23)(22,28)(24,32)(25,34)(26,33)(27,35)(29,41)(30,43)(31,44)(36,47)(37,49)(38,50)(39,42)(40,48)(45,51)(46,52);(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24)(22,29)(23,32)(25,37)(26,40)(27,38)(28,41)(30,45)(31,46)(33,48)(34,49)(35,50)(36,39)(42,47)(43,51)(44,52);(8,16)(11,20)(13,15)(17,19)(21,25,27,26)(22,30,31,39)(23,33,35,34)(24,37,38,40)(28,42,44,43)(29,45,46,36)(32,48,50,49)(41,47,52,51);(7,14)(10,18)(13,15)(17,19)(21,25,27,26)(22,30,31,39)(23,34,35,33)(24,37,38,40)(28,43,44,42)(29,45,46,36)(32,49,50,48)(41,51,52,47)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6415940);