Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2431 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1588 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)(3,7)(4,8)(5,6)(9,10,12,13)(11,15,16,14)(17,18,23,28)(19,30,33,25)(20,34,32,42)(21,36,39,45)(22,29,24,27)(26,40,44,38)(31,43,47,48)(35,46,37,41);(2,6)(7,8)(10,13)(11,16)(17,19)(18,25)(20,24)(21,37)(22,32)(23,33)(26,43)(27,42)(28,30)(29,34)(31,40)(35,39)(36,46)(38,47)(41,45)(44,48);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20,22,33,23,32,24,19)(18,25,27,34,28,30,29,42)(21,35,38,47,39,37,40,31)(26,41,36,48,44,46,45,43);(1,4,5,3)(2,8,6,7)(10,13)(14,15)(17,21,24,40,23,39,22,38)(18,26,29,45,28,44,27,36)(19,31,32,37,33,47,20,35)(25,41,42,43,30,46,34,48);(2,6)(7,8)(10,13)(14,15)(17,22,23,24)(18,27,28,29)(19,32,33,20)(21,38,39,40)(25,42,30,34)(26,36,44,45)(31,37,47,35)(41,43,46,48)Polycyclic presentation:
2^9|a^2=ghi,c^2=fh,d^2=fg,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7510534);