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 | |
7235 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4)(5,6,10,13)(7,14,15,11)(8,9,12,19)(16,17,18,20)(22,26)(27,29)(28,32)(35,36);(1,2)(3,4)(5,7)(6,11)(8,18)(9,17)(10,15)(12,16)(13,14)(19,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,36)(32,34)(33,35);(7,15)(11,14)(16,18)(17,20)(21,22,25,26)(23,27,31,29)(24,32,34,28)(30,36,33,35);(5,8)(6,9)(7,16)(10,12)(11,20)(13,19)(14,17)(15,18)(21,23)(22,27)(24,33)(25,31)(26,29)(28,36)(30,34)(32,35);(5,7)(6,11)(8,18)(9,17)(10,15)(12,16)(13,14)(19,20)(21,22)(23,27)(24,28)(25,26)(29,31)(30,35)(32,34)(33,36);(5,9)(6,12)(7,17)(8,13)(10,19)(11,16)(14,18)(15,20)(21,24)(22,28)(23,30)(25,34)(26,32)(27,35)(29,36)(31,33)Polycyclic presentation:
2^9|a^2=gh,c^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72336756 |
| maximal normal subgroup of | #72336756 |
| maximal quotient of | #72336919 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7984597);