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 | #61270 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1600 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)(19,29)(20,26)(21,38)(22,39)(23,27)(24,33)(25,37)(28,40)(30,42)(31,46)(32,43)(34,44)(35,41)(36,45)(47,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,24)(20,35)(21,30)(22,34)(23,32)(25,42)(26,46)(27,44)(28,43)(29,39)(31,37)(33,40)(36,48)(38,47)(41,45);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20)(18,25)(19,30)(21,23)(22,36)(24,41)(26,39)(27,45)(28,38)(29,47)(31,34)(32,35)(33,48)(37,40)(42,44)(43,46);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,21,23,20)(18,26,27,38)(19,31,32,48)(22,36,40,37)(24,42,44,41)(25,28,45,39)(29,47,43,46)(30,33,35,34);(2,8)(6,9)(7,16)(14,15)(18,27)(19,32)(25,45)(26,38)(28,39)(30,35)(31,48)(33,34)Polycyclic presentation:
2^9|c^2=hi,d^2=h,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, 7031310);