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 | |
7200 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)(7,8)(9,14)(10,12)(11,13)(15,16)(17,19)(18,20)(21,22)(23,27)(24,32)(25,30)(26,28)(29,34)(31,36)(33,35);(1,2)(3,4)(5,7)(6,11)(8,12)(9,17)(10,13)(14,16)(15,18)(19,20)(21,23,26,25)(22,27,28,30)(24,31,34,33)(29,35,32,36);(6,12)(7,13)(9,18)(16,19)(22,28)(27,30)(29,32)(35,36);(5,7)(6,8)(9,15)(10,13)(11,12)(14,16)(17,18)(19,20)(21,22)(23,27)(24,29)(25,30)(26,28)(31,35)(32,34)(33,36);(5,8)(6,13)(7,12)(9,16)(10,11)(14,17)(15,20)(18,19)(21,24)(22,29)(23,31)(25,33)(26,34)(27,35)(28,32)(30,36);(5,9,10,18)(6,14,12,20)(7,15,13,17)(8,16,11,19)(21,25,26,23)(22,30,28,27)(24,33,34,31)(29,36,32,35)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72336756 |
| maximal normal subgroup of | #72336756 |
| maximal quotient of | #72336893, #72336915 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7731586);