Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6631116 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2615 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26)(27,38)(28,35)(29,45)(30,36)(31,37)(32,47)(33,42)(34,44)(39,53)(40,49)(41,54)(43,50)(46,52)(48,56)(51,55);(3,7)(5,8)(10,14)(15,20)(16,21)(22,24)(25,27,31,42)(26,33,37,38)(28,40,29,39)(30,41,32,43)(34,51,52,48)(35,53,45,49)(36,50,47,54)(44,56,46,55);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,31,29)(26,34,37,52)(27,39,42,40)(30,44,32,46)(33,48,38,51)(35,36,45,47)(41,55,43,56)(49,50,53,54);(2,6)(3,7)(11,17)(15,20)(18,23)(22,24)(25,29,31,28)(26,35,37,45)(27,40,42,39)(30,44,32,46)(33,49,38,53)(34,36,52,47)(41,55,43,56)(48,50,51,54);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,30,31,32)(26,36,37,47)(27,41,42,43)(28,44,29,46)(33,50,38,54)(34,45,52,35)(39,55,40,56)(48,53,51,49)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^b=di,d^c=dfh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538635);