Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437969 |
| 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 | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,10,14,11,15,9,21)(7,12,17,26,18,27,23,25)(8,13,19,29,20,30,22,28)(16,24,32,34,33,35,31,36);(1,2)(3,4)(5,6)(7,12)(8,13)(9,14)(10,21)(11,15)(16,24)(17,25)(18,27)(19,28)(20,30)(22,29)(23,26)(31,34)(32,36)(33,35);(2,4)(5,7,11,18)(6,12,15,27)(8,16,20,33)(9,17,10,23)(13,24,30,35)(14,25,21,26)(19,31,22,32)(28,34,29,36);(2,4)(5,8)(6,13)(7,16)(9,22)(10,19)(11,20)(12,24)(14,29)(15,30)(17,32)(18,33)(21,28)(23,31)(25,36)(26,34)(27,35);(5,9,11,10)(6,14,15,21)(7,17,18,23)(8,19,20,22)(12,25,27,26)(13,28,30,29)(16,31,33,32)(24,34,35,36)Polycyclic presentation:
2^9|a^2=f,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,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 | #72318090, #72318152, #72318287, … |
| maximal normal subgroup of | #72318090, #72318152, #72318287, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345175);