Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,15)(11,12,17,20)(13,16,21,22)(18,19,23,24)(25,26,30,32)(27,33,34,31)(28,29,37,38)(35,39,40,36);(1,2,5,6)(3,7,4,8)(25,27,30,34)(26,31,32,33)(28,35,37,40)(29,36,38,39);(10,15)(11,17)(16,22)(18,23)(26,32)(27,34)(28,37)(36,39);(1,3,5,4)(2,7,6,8)(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,23)(22,24)(25,28)(26,29)(27,35)(30,37)(31,36)(32,38)(33,39)(34,40);(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,27)(26,31)(28,35)(29,36)(30,34)(32,33)(37,40)(38,39);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,19,20,24)(25,29)(26,28)(27,36)(30,38)(31,35)(32,37)(33,40)(34,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=gi,f^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7624924);