Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
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 | |
6820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4)(5,6,11,12)(7,9,8,13)(10,14,18,19)(15,17,16,20)(21,22)(23,27)(24,28)(25,29)(26,30)(31,35)(32,36)(33,34);(1,2)(3,4)(5,7)(6,9)(8,11)(10,15)(12,13)(14,17)(16,18)(19,20)(21,22,26,30)(23,28,24,27)(25,29,34,33)(31,36,32,35);(6,12)(7,8)(10,18)(17,20)(21,22)(23,27)(24,28)(25,33)(26,30)(29,34)(31,36)(32,35);(5,8,11,7)(6,13,12,9)(10,16,18,15)(14,20,19,17)(21,23,26,24)(22,27,30,28)(25,31,34,32)(29,35,33,36);(5,9,11,13)(6,7,12,8)(10,17,18,20)(14,15,19,16)(21,24,26,23)(22,28,30,27)(25,32,34,31)(29,36,33,35);(5,10,11,18)(6,14,12,19)(7,15,8,16)(9,17,13,20)(21,25,26,34)(22,29,30,33)(23,31,24,32)(27,35,28,36)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,d^2=i,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7670614);