Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573715 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,10,14)(7,11,17,24)(8,12,19,26)(9,20,21,13)(15,27,29,22)(16,30,31,23)(18,32,33,25)(28,34,36,35);(1,2)(3,4)(5,7)(6,11)(8,15)(9,16)(10,17)(12,22)(13,23)(14,24)(18,28)(19,29)(20,30)(21,31)(25,34)(26,27)(32,35)(33,36);(2,4)(5,8)(6,12)(7,15)(9,18)(10,19)(11,22)(13,25)(14,26)(16,28)(17,29)(20,32)(21,33)(23,34)(24,27)(30,35)(31,36);(1,2)(3,4)(5,9,10,21)(6,13,14,20)(7,16,17,31)(8,18,19,33)(11,23,24,30)(12,25,26,32)(15,28,29,36)(22,34,27,35);(2,4)(6,14)(11,24)(12,26)(13,20)(22,27)(23,30)(25,32)(34,35)Polycyclic presentation:
2^9|a^2=i,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72267070, #72267071, #72267074, … |
| maximal normal subgroup of | #72267070, #72267071, #72267074, … |
| maximal quotient of | #72267126, #72267145 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480923);