Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996405 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #61261 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1587 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,10,14)(4,7,12,16)(9,13,11,15)(17,18,22,26,23,20,34,27)(19,28,32,29,33,40,31,24)(21,25,37,42,38,35,45,43)(30,41,36,44,47,48,46,39);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19,23,33)(18,24,20,29)(21,36,38,46)(22,31,34,32)(25,41,35,48)(26,40,27,28)(30,45,47,37)(39,43,44,42);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,20)(18,23)(19,29)(21,25)(22,27)(24,33)(26,34)(28,31)(30,39)(32,40)(35,38)(36,41)(37,42)(43,45)(44,47)(46,48);(4,12)(7,16)(9,11)(13,15)(17,21,34,45,23,38,22,37)(18,25,27,43,20,35,26,42)(19,30,31,46,33,47,32,36)(24,39,40,48,29,44,28,41);(2,8)(3,10)(7,16)(9,11)(17,22,23,34)(18,26,20,27)(19,31,33,32)(21,37,38,45)(24,40,29,28)(25,42,35,43)(30,46,47,36)(39,48,44,41)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=h,d^2=fi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7011612);