Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2365 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)(13,14,18,22,19,23,17,34)(15,24,28,37,29,26,27,20)(16,21,32,40,33,31,35,39)(25,41,42,38,30,43,44,36);(5,6)(7,11)(8,10)(9,12)(13,15,19,29)(14,20,23,37)(16,30,33,25)(17,28,18,27)(21,38,31,36)(22,26,34,24)(32,42,35,44)(39,43,40,41);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,25)(17,35)(18,32)(19,33)(20,36)(22,40)(23,31)(24,41)(26,43)(27,44)(28,42)(29,30)(34,39)(37,38);(7,12)(9,11)(13,14,17,34,19,23,18,22)(15,26,27,37,29,24,28,20)(16,31,35,40,33,21,32,39)(25,41,44,36,30,43,42,38);(6,10)(9,11)(13,17,19,18)(14,22,23,34)(15,27,29,28)(16,32,33,35)(20,24,37,26)(21,39,31,40)(25,42,30,44)(36,43,38,41)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6903974);