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 | |
4405 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,20)(21,24)(22,23)(25,26,31,29)(27,28,35,32)(30,33,39,37)(34,36,40,38);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,17)(18,19)(23,24)(25,27)(26,28)(29,32)(30,34)(31,35)(33,36)(37,38)(39,40);(11,20)(13,21)(15,19)(17,24)(25,28,31,32)(26,27,29,35)(30,38,39,36)(33,40,37,34);(9,10)(11,15)(12,23)(13,17)(14,18)(16,22)(19,20)(21,24)(25,29,31,26)(27,32,35,28)(30,37,39,33)(34,38,40,36);(1,3,5,4)(2,7,6,8)(9,12,14,22)(10,16,18,23)(11,21,20,13)(15,17,19,24)(25,27)(26,32)(28,29)(30,34)(31,35)(33,38)(36,37)(39,40);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,17,18,24)(11,22,20,12)(15,16,19,23)(25,30,31,39)(26,33,29,37)(27,34,35,40)(28,36,32,38)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=h,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7675271);