Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,14)(8,17)(9,19)(10,12)(11,16)(13,20)(15,18)(21,22)(23,26)(24,27)(25,28)(29,35)(30,33)(31,34)(32,36);(1,3,4,2)(5,7)(6,11)(8,12)(9,17)(10,15)(13,18)(14,19)(16,20)(21,23)(22,26)(24,29)(25,30)(27,32)(28,33)(31,36)(34,35);(5,8,10,11)(6,7,12,15)(9,18,20,14)(13,16,19,17)(21,24)(22,27)(23,29)(25,31)(26,32)(28,34)(30,36)(33,35);(6,12)(7,15)(13,19)(16,17)(22,28)(26,33)(27,34)(32,35)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,d^b=dh,e^b=eh,f^b=fi,e^c=ei,f^c=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 99598);