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 | #45316 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1986 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)(3,4)(7,11)(10,12)(13,14,19,25)(15,21,28,37)(16,22,31,40)(17,24,18,23)(20,32,35,27)(26,36,43,38)(29,44,42,34)(30,41,33,39);(6,9)(7,11)(13,15)(14,20)(16,29)(17,27)(18,32)(19,28)(21,23)(22,38)(24,37)(25,35)(26,30)(31,42)(33,43)(34,39)(36,40)(41,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,26)(17,32)(18,27)(19,28)(20,23)(22,36)(24,35)(25,37)(29,30)(31,43)(33,42)(34,39)(38,40)(41,44);(6,9)(10,12)(13,16,19,31)(14,22,25,40)(15,26,28,43)(17,30,18,33)(20,34,35,44)(21,36,37,38)(23,39,24,41)(27,42,32,29);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,23,25,24)(15,27,28,32)(16,30,31,33)(20,21,35,37)(22,39,40,41)(26,42,43,29)(34,36,44,38)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7050362);