Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | #42883 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,4,3)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22)(23,28)(24,34)(25,30)(26,39)(27,31)(29,32)(33,42)(35,47)(36,41)(37,45)(38,49)(40,44)(43,46)(48,51)(50,52);(2,3)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,26)(22,28,31,39)(24,32,36,40)(25,33,37,38)(29,41,44,34)(30,42,45,49)(35,46,48,50)(43,51,52,47);(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24,27,36)(22,29,31,44)(23,32,26,40)(25,35,37,48)(28,41,39,34)(30,43,45,52)(33,46,38,50)(42,51,49,47);(1,3,4,2)(8,16)(11,20)(13,15)(17,19)(21,25)(22,30)(23,33)(24,35)(26,38)(27,37)(28,42)(29,43)(31,45)(32,46)(34,47)(36,48)(39,49)(40,50)(41,51)(44,52);(6,12)(7,14)(11,20)(13,15)(25,37)(30,45)(33,38)(35,48)(42,49)(43,52)(46,50)(47,51)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6941915);