Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1709 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)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,27,31)(23,28,34,44)(24,36,39,46)(25,30,26,40)(29,41,47,38)(32,48,37,52)(33,43,35,50)(42,51,45,49);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23,27,34)(22,28,31,44)(24,37,39,32)(25,33,26,35)(29,45,47,42)(30,43,40,50)(36,52,46,48)(38,51,41,49);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,24)(22,29)(23,32)(25,38)(26,41)(27,39)(28,42)(30,46)(31,47)(33,49)(34,37)(35,51)(36,40)(43,52)(44,45)(48,50);(7,15)(10,13)(14,20)(17,19)(21,25,27,26)(22,30,31,40)(23,33,34,35)(24,38,39,41)(28,43,44,50)(29,46,47,36)(32,49,37,51)(42,52,45,48);(1,3,4,2)(6,12)(10,13)(11,18)(17,19)(23,34)(24,39)(28,44)(29,47)(33,35)(36,46)(38,41)(43,50)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=f,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^b=ei,e^c=ei,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6306565);