Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885065 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1643 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,23,28,38)(20,30,32,42)(21,34,36,25)(24,35,41,33)(27,43,31,48)(29,46,45,39)(37,47,40,44);(1,3,7,4)(2,5,8,6)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,31)(21,29)(22,28)(24,40)(25,39)(26,38)(27,32)(30,48)(33,47)(34,46)(35,44)(36,45)(37,41)(42,43);(1,4,7,3)(2,5,8,6)(11,16)(13,15)(17,20)(18,24)(19,27)(21,35)(22,32)(23,37)(25,42)(26,41)(28,31)(29,47)(30,34)(33,36)(38,40)(39,48)(43,46)(44,45);(1,5,7,6)(2,3,8,4)(10,14)(13,15)(19,28)(20,32)(23,38)(24,41)(29,45)(30,42)(33,35)(39,46);(1,6,7,5)(2,3,8,4)(19,28)(23,38)(27,31)(29,45)(37,40)(39,46)(43,48)(44,47)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fhi,d^2=fhi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6900104);