Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993887 |
| 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 | #3493 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1568 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)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,18,24,29)(19,30,25,35)(20,32,36,33)(21,37,41,26)(22,28,23,27)(31,44,43,47)(34,48,39,38)(40,45,42,46);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,20)(21,38)(22,33)(23,32)(24,25)(26,44)(27,30)(28,35)(29,36)(31,40)(34,45)(37,47)(39,46)(41,48)(42,43);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20)(18,25)(19,29)(21,39)(22,35)(23,30)(24,36)(26,31)(27,33)(28,32)(34,41)(37,43)(38,45)(40,47)(42,44)(46,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,21,24,41)(18,26,29,37)(19,31,25,43)(20,34,36,39)(22,40,23,42)(27,45,28,46)(30,47,35,44)(32,38,33,48);(2,8)(6,9)(7,16)(14,15)(17,22,24,23)(18,27,29,28)(19,32,25,33)(20,35,36,30)(21,40,41,42)(26,45,37,46)(31,38,43,48)(34,44,39,47)Polycyclic presentation:
2^9|a^2=i,c^2=hi,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,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=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7031317);