Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432843 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20)(21,22)(23,28)(24,30)(25,34)(26,29)(27,31)(32,36)(33,35);(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(21,23,26,32)(22,27,29,35)(24,31,25,33)(28,30,36,34);(2,4)(7,13)(10,16)(14,19)(17,20)(21,23)(22,28)(24,33)(25,31)(26,32)(27,34)(29,36)(30,35);(6,11)(10,16)(12,18)(17,20)(22,29)(27,35)(28,36)(30,34);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)(21,24,26,25)(22,30,29,34)(23,31,32,33)(27,36,35,28)Polycyclic presentation:
2^9|b^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72304664, #72304713, #72304778, … |
| maximal normal subgroup of | #72304664, #72304713, #72304778, … |
| maximal quotient of | #72305037, #72305067 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343747);