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 | #31068 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1873 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)(13,14,19,21,18,24,17,23)(15,20,26,33,28,36,27,35)(16,29,30,38,32,37,31,22)(25,39,42,44,41,43,40,34);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,20)(16,25)(17,27)(18,28)(19,26)(21,33)(22,34)(23,35)(24,36)(29,39)(30,42)(31,40)(32,41)(37,43)(38,44);(2,4)(14,21)(17,19)(20,33)(22,37)(23,24)(26,27)(29,38)(30,31)(34,43)(35,36)(39,44)(40,42);(6,11)(7,12)(13,16,18,32)(14,22,24,38)(15,25,28,41)(17,30,19,31)(20,34,36,44)(21,37,23,29)(26,40,27,42)(33,43,35,39);(5,8,9,10)(6,7,11,12)(13,17,18,19)(14,21,24,23)(15,26,28,27)(16,30,32,31)(20,35,36,33)(22,37,38,29)(25,40,41,42)(34,39,44,43)Polycyclic presentation:
2^9|a^2=gi,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, 7411238);