Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437452 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6084 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,16)(8,13)(9,14)(10,22)(11,15)(12,18)(17,32)(19,26)(20,28)(21,30)(23,27)(24,29)(25,34)(31,35)(33,36);(1,2)(3,4)(5,7)(6,12)(8,17)(9,18)(10,23)(11,19)(13,25)(14,26)(15,27)(16,22)(20,33)(21,35)(24,34)(28,32)(29,31)(30,36);(2,4)(5,8,11,21)(6,13,15,30)(7,17,19,35)(9,20,10,24)(12,25,27,36)(14,28,22,29)(16,31,26,32)(18,33,23,34);(2,4)(5,6)(7,12)(8,13)(9,22)(10,14)(11,15)(16,18)(17,25)(19,27)(20,29)(21,30)(23,26)(24,28)(31,33)(32,34)(35,36);(5,9,11,10)(6,14,15,22)(7,18,19,23)(8,20,21,24)(12,26,27,16)(13,28,30,29)(17,33,35,34)(25,32,36,31)Polycyclic presentation:
2^9|c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72302921, #72303071, #72303154, … |
| maximal normal subgroup of | #72302921, #72303071, #72303154, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344654);