Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #42679 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2169 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(7,11)(10,12)(13,14,18,22)(15,19,24,33)(16,27,28,21)(17,20,30,23)(25,39,37,32)(26,34,38,31)(29,36,41,35)(40,42,44,43);(2,3)(6,9)(7,11)(13,15)(14,19)(16,25)(17,26)(18,24)(20,34)(21,32)(22,33)(23,31)(27,39)(28,37)(29,40)(30,38)(35,43)(36,42)(41,44);(5,6)(7,10)(8,9)(11,12)(14,20)(15,24)(16,28)(17,30)(19,31)(21,36)(22,23)(27,35)(32,43)(33,34)(39,42)(40,44);(1,3,4,2)(6,9)(10,12)(13,16,18,28)(14,21,22,27)(15,25,24,37)(17,29,30,41)(19,32,33,39)(20,35,23,36)(26,40,38,44)(31,42,34,43);(5,7,8,11)(6,10,9,12)(14,22)(15,24)(16,28)(20,23)(26,38)(29,41)(32,39)(42,43)Polycyclic presentation:
2^9|a^2=fi,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7053150);