Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | C2×#3322 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1904 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,24)(15,21,25,36)(16,22,30,37)(17,23,18,20)(26,35,39,44)(27,32,31,34)(28,33,29,38)(40,43,41,42);(5,6)(7,11)(8,10)(9,12)(14,20)(15,25)(16,28)(17,18)(21,34)(23,24)(26,40)(29,30)(32,36)(33,38)(35,44)(39,41);(2,4)(5,7,8,12)(6,9,10,11)(13,15)(14,21)(16,26)(17,31)(18,27)(19,25)(20,32)(22,35)(23,34)(24,36)(28,41)(29,40)(30,39)(33,42)(37,44)(38,43);(7,12)(9,11)(13,16,19,30)(14,22,24,37)(15,26,25,39)(17,29,18,28)(20,33,23,38)(21,35,36,44)(27,41,31,40)(32,42,34,43);(6,10)(9,11)(13,17,19,18)(14,20,24,23)(15,27,25,31)(16,29,30,28)(21,34,36,32)(22,33,37,38)(26,41,39,40)(35,43,44,42)Polycyclic presentation:
2^9|a^2=gi,c^2=hi,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6970210);