Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432861 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1651 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,29)(20,26)(21,27)(22,24)(23,28)(25,31)(30,43)(32,39)(33,42)(34,37)(35,41)(36,40)(38,46)(44,47)(45,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(18,24)(20,33)(21,22)(25,29)(26,41)(27,28)(30,44)(31,36)(37,42)(38,47)(39,40)(43,48);(11,15)(13,16)(17,19,23,32)(18,25,28,40)(20,30,33,44)(21,31,22,36)(24,29,27,39)(26,38,42,48)(34,45,35,46)(37,43,41,47);(10,14)(13,16)(17,20)(18,26)(19,30)(21,35)(22,34)(23,33)(24,37)(25,38)(27,41)(28,42)(29,43)(31,46)(32,44)(36,45)(39,47)(40,48);(1,3,4,7)(2,6,5,8)(17,21,23,22)(18,27,28,24)(19,31,32,36)(20,34,33,35)(25,39,40,29)(26,37,42,41)(30,45,44,46)(38,43,48,47)Polycyclic presentation:
2^9|c^2=i,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341741);