Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500489 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60565 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1626 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)(3,6,9,12)(4,7,11,14)(10,13,15,16)(18,24)(19,28)(21,22)(25,38)(26,37)(27,35)(29,41)(30,31)(33,34)(39,47)(40,45)(42,43);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,18)(19,25)(20,26)(21,35)(22,24)(23,27)(28,36)(29,39)(30,44)(31,38)(32,40)(33,45)(34,37)(41,46)(42,47)(43,48);(3,9)(4,11)(6,12)(7,14)(17,19,22,28,23,31,21,30)(18,25,24,36,27,38,35,44)(20,29,34,41,32,42,33,43)(26,39,37,46,40,47,45,48);(1,4,5,11)(2,7,8,14)(3,10,9,15)(6,13,12,16)(17,20)(18,26)(19,29)(21,33)(22,34)(23,32)(24,37)(25,39)(27,40)(28,41)(30,43)(31,42)(35,45)(36,46)(38,47)(44,48);(3,9)(6,12)(10,15)(13,16)(20,32)(26,40)(29,42)(33,34)(37,45)(39,47)(41,43)(46,48)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6434019);