Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1905 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22)(23,30)(24,29)(25,27)(26,33)(28,31)(32,36)(34,35);(2,3)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(22,27)(28,35)(29,33)(30,36);(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,23)(22,28)(24,31)(25,32)(26,34)(27,35)(29,36)(30,33);(1,3,4,2)(8,16)(11,20)(13,15)(17,19)(21,22)(23,28)(24,33)(25,27)(26,29)(30,31)(32,35)(34,36);(6,12)(7,14)(11,20)(13,15)(21,24,25,26)(22,29,27,33)(23,31,32,34)(28,36,35,30)Polycyclic presentation:
2^9|a^2=fhi,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=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6941922);