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×#3269 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2107 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)(5,6)(7,9)(8,10)(11,12)(13,14,19,24)(15,21,27,36)(16,22,30,38)(17,23,18,20)(25,34,41,43)(26,32,31,35)(28,33,29,37)(39,42,40,44);(5,6)(7,11)(8,10)(9,12)(14,20)(16,28)(17,18)(21,32)(23,24)(25,39)(26,31)(29,30)(33,37)(35,36)(40,41)(42,44);(2,4)(5,7,8,12)(6,9,10,11)(13,15,19,27)(14,21,24,36)(16,25,30,41)(17,31,18,26)(20,32,23,35)(22,34,38,43)(28,39,29,40)(33,42,37,44);(7,12)(9,11)(13,16)(14,22)(15,25)(17,29)(18,28)(19,30)(20,33)(21,34)(23,37)(24,38)(26,39)(27,41)(31,40)(32,42)(35,44)(36,43);(6,10)(9,11)(13,17,19,18)(14,23,24,20)(15,26,27,31)(16,29,30,28)(21,32,36,35)(22,37,38,33)(25,39,41,40)(34,42,43,44)Polycyclic presentation:
2^9|a^2=i,c^2=hi,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=ehi,f^b=fi,d^c=dh,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, 6961964);