Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #45412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(7,8)(9,10,12,13)(11,14,16,15)(17,18,22,26)(19,27,28,40)(20,24,36,44)(21,37,29,25)(23,31,38,32)(30,45,33,48)(34,43,35,42)(39,47,41,46);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(19,28)(23,38)(27,40)(30,33)(31,32)(39,41)(45,48)(46,47);(1,3,5,4)(2,7,6,8)(10,13)(11,16)(17,19,29,31,22,28,21,32)(18,23,37,40,26,38,25,27)(20,33,34,46,36,30,35,47)(24,41,42,45,44,39,43,48);(1,4,5,3)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20,22,36)(18,24,26,44)(19,30,28,33)(21,35,29,34)(23,39,38,41)(25,43,37,42)(27,45,40,48)(31,47,32,46);(10,13)(14,15)(17,21,22,29)(18,25,26,37)(19,31,28,32)(20,34,36,35)(23,40,38,27)(24,42,44,43)(30,46,33,47)(39,45,41,48)Polycyclic presentation:
2^9|a^2=h,c^2=fgi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6951070);