Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(6,9)(7,11)(13,14,19,24)(15,20,28,36)(16,22,30,37)(17,23,18,32)(21,31,38,29)(25,34,40,43)(26,35,27,42)(33,41,44,39);(2,3)(5,6)(7,10)(8,9)(11,12)(13,15,19,28)(14,20,24,36)(16,25,30,40)(17,26,18,27)(21,33,38,44)(22,34,37,43)(23,35,32,42)(29,39,31,41);(1,2,4,3)(13,16)(14,21)(15,25)(17,29)(18,31)(19,30)(20,33)(22,32)(23,37)(24,38)(26,39)(27,41)(28,40)(34,42)(35,43)(36,44);(5,7,8,11)(6,10,9,12)(13,16)(14,22)(15,25)(17,31)(18,29)(19,30)(20,34)(21,32)(23,38)(24,37)(26,41)(27,39)(28,40)(33,42)(35,44)(36,43);(1,3,4,2)(6,9)(10,12)(13,17,19,18)(14,23,24,32)(15,26,28,27)(16,29,30,31)(20,35,36,42)(21,37,38,22)(25,39,40,41)(33,43,44,34)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,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, 6715484);