Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
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 | C2×#537 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,3,4)(5,6)(7,13)(8,12)(9,11)(10,14)(15,19)(16,18)(17,20)(21,22)(23,28)(24,35)(25,30)(26,39)(27,31)(29,36)(32,47)(33,42)(34,43)(37,45)(38,44)(40,46)(41,50)(48,51)(49,52);(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19)(21,23)(22,28)(24,32)(25,38)(26,33)(27,34)(29,41)(30,44)(31,43)(35,47)(36,50)(37,49)(39,42)(40,48)(45,52)(46,51);(7,14)(10,13)(15,20)(17,19)(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)(43,51)(44,52);(6,11)(10,13)(12,18)(17,19)(21,25,27,26)(22,30,31,39)(23,33,34,38)(24,36,37,40)(28,42,43,44)(29,45,46,35)(32,48,49,50)(41,47,51,52);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)(23,34)(28,43)(32,49)(33,38)(41,51)(42,44)(47,52)(48,50)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,d^b=di,e^b=ei,g^c=gi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281881);