Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| 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 | #61274 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1579 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)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,18,22,26)(19,28,33,43)(20,34,37,25)(21,38,39,27)(23,32,42,31)(24,35,44,36)(29,41,48,40)(30,45,47,46);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,23)(20,29)(21,32)(22,33)(24,40)(25,45)(26,42)(27,43)(28,38)(30,35)(31,39)(34,46)(36,47)(37,48)(41,44);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20)(18,24)(19,29)(21,35)(22,37)(23,40)(25,27)(26,44)(28,46)(30,32)(31,47)(33,48)(34,38)(36,39)(41,42)(43,45);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20)(18,25)(19,30)(21,36)(22,37)(23,41)(24,27)(26,34)(28,46)(29,32)(31,48)(33,47)(35,39)(38,44)(40,42)(43,45);(2,8)(6,9)(7,16)(14,15)(18,26)(23,42)(24,44)(25,34)(27,38)(28,43)(40,41)(45,46)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7031328);