Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021351 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
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)(17,18,23,27,24,28,22,39)(19,29,33,43,34,42,32,25)(20,35,36,45,37,47,21,26)(30,40,46,41,31,44,38,48);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19)(18,25)(20,31)(21,38)(22,33)(23,32)(24,34)(26,44)(27,42)(28,43)(29,39)(30,37)(35,41)(36,46)(40,45)(47,48);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,20,22,21,24,37,23,36)(18,26,27,35,28,45,39,47)(19,30,33,46,34,31,32,38)(25,40,42,48,43,44,29,41);(4,12)(7,16)(9,11)(13,15)(17,21,22,37,24,36,23,20)(18,26,39,47,28,45,27,35)(19,31,32,46,34,30,33,38)(25,41,42,40,43,48,29,44);(2,8)(3,10)(7,16)(9,11)(17,22,24,23)(18,27,28,39)(19,32,34,33)(20,21,37,36)(25,29,43,42)(26,35,45,47)(30,38,31,46)(40,41,44,48)Polycyclic presentation:
2^9|a^2=fhi,c^2=g,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=dh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949274);