Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| 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 |
| prime radical | #31207 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1867 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,8)(9,11)(10,12)(13,14,20,24,19,26,18,25)(15,21,28,23,17,34,29,37)(16,30,35,22,33,39,32,40)(27,38,41,36,31,43,42,44);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,19,17)(14,21,26,34)(16,27,33,31)(18,29,20,28)(22,36,40,44)(23,25,37,24)(30,38,39,43)(32,42,35,41);(2,4)(13,16,19,33)(14,22,26,40)(15,27,17,31)(18,35,20,32)(21,36,34,44)(23,38,37,43)(24,30,25,39)(28,42,29,41);(6,11)(7,12)(13,17,19,15)(14,23,26,37)(16,31,33,27)(18,29,20,28)(21,25,34,24)(22,38,40,43)(30,36,39,44)(32,42,35,41);(5,8,9,10)(6,7,11,12)(13,18,19,20)(14,24,26,25)(15,28,17,29)(16,32,33,35)(21,37,34,23)(22,39,40,30)(27,41,31,42)(36,38,44,43)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411240);