Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514144 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #58990 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3089 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,13)(4,18)(5,7)(6,12)(8,16)(9,20)(10,19)(11,22)(14,26)(15,23)(17,25)(21,27)(24,31)(28,29)(30,32);(2,7)(3,14)(5,22)(9,27)(10,18)(11,12)(13,25)(15,30)(16,17)(19,20)(28,32)(29,31);(1,3)(2,8)(4,15)(5,14)(6,17)(7,13)(9,23)(10,24)(11,25)(12,26)(16,22)(18,28)(19,29)(20,30)(21,31)(27,32);(1,4)(2,9)(3,15)(5,19)(6,21)(7,18)(8,23)(10,12)(11,27)(13,28)(14,29)(16,30)(17,31)(20,22)(24,26)(25,32);(1,4)(2,10)(3,15)(5,20)(6,21)(7,18)(8,24)(9,12)(11,27)(13,28)(14,30)(16,29)(17,31)(19,22)(23,26)(25,32)Polycyclic presentation:
2^9|f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,f^b=fi,g^b=gi,g^c=gi,h^c=hi,g^d=gi,f^e=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72214082, #72226279, #72226285 |
| maximal normal subgroup of | #72214082 |
| maximal quotient of | #72318957, #72341442, #72341496 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421340);