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 | #42673 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
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)(23,27)(24,29)(25,28)(26,31)(30,37)(32,39)(33,38)(34,41)(35,43)(36,42)(40,47)(44,48)(45,50)(46,49)(51,52);(2,3)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,26,34)(22,27,31,41)(24,32,35,45)(25,33,37,47)(28,38,30,40)(29,39,43,50)(36,46,48,52)(42,49,44,51);(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(22,28)(25,37)(27,38)(29,42)(30,31)(33,47)(36,48)(39,49)(40,41)(43,44)(46,52)(50,51);(1,3,4,2)(8,16)(11,20)(13,15)(17,19)(21,24)(22,29)(23,32)(25,36)(26,35)(27,39)(28,42)(30,44)(31,43)(33,46)(34,45)(37,48)(38,49)(40,51)(41,50)(47,52);(6,12)(7,14)(11,20)(13,15)(24,35)(29,43)(32,45)(36,48)(39,50)(42,44)(46,52)(49,51)Polycyclic presentation:
2^9|a^2=fh,b^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, 6941911);