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 | #45412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1829 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)(5,6)(7,9)(8,10)(11,12)(13,14,20,33,19,25,18,24)(15,21,34,43,28,38,27,37)(16,29,30,22,17,23,31,39)(26,36,41,35,32,40,42,44);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15)(14,21)(16,26)(17,32)(18,27)(19,28)(20,34)(22,35)(23,40)(24,37)(25,38)(29,36)(30,41)(31,42)(33,43)(39,44);(5,7,8,12)(6,9,10,11)(13,16,19,17)(14,22,25,39)(15,26,28,32)(18,30,20,31)(21,35,38,44)(23,33,29,24)(27,41,34,42)(36,37,40,43);(1,3,4,2)(7,12)(9,11)(13,17,19,16)(14,23,25,29)(15,26,28,32)(18,30,20,31)(21,36,38,40)(22,33,39,24)(27,42,34,41)(35,37,44,43);(6,10)(9,11)(13,18,19,20)(14,24,25,33)(15,27,28,34)(16,30,17,31)(21,37,38,43)(22,23,39,29)(26,41,32,42)(35,40,44,36)Polycyclic presentation:
2^9|a^2=fgi,c^2=hi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,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, 6971554);