Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431795 |
| 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 | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
7456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,11)(8,12)(9,13)(10,14)(15,21)(16,22)(17,23)(18,25)(19,26)(20,27)(24,29)(28,32)(30,33)(31,35)(34,36);(5,7,10,17)(6,11,14,23)(8,15,19,29)(9,16,20,30)(12,24,26,21)(13,22,27,33)(18,28,31,36)(25,34,35,32);(2,4)(5,8)(6,12)(7,15)(9,18)(10,19)(11,21)(13,25)(14,26)(16,28)(17,29)(20,31)(22,32)(23,24)(27,35)(30,36)(33,34);(5,9)(6,13)(7,16)(8,18)(10,20)(11,22)(12,25)(14,27)(15,28)(17,30)(19,31)(21,32)(23,33)(24,34)(26,35)(29,36);(9,20)(13,27)(16,30)(18,31)(22,33)(25,35)(28,36)(32,34)Polycyclic presentation:
2^9|b^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72240660, #72251087, #72251117, … |
| maximal normal subgroup of | #72240660, #72251087, #72251117, … |
| maximal quotient of | #72238065, #72251254, #72251632, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338992);