Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632108 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #36702 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2219 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)(3,11)(4,9)(5,10)(6,12)(7,13)(8,15)(14,29)(16,24)(17,23)(18,32)(19,26)(20,28)(21,27)(22,31)(25,30);(1,3)(2,8)(4,18)(5,17)(6,15)(7,16)(9,25)(10,11)(12,24)(13,23)(14,19)(20,31)(21,30)(22,27)(26,32)(28,29);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,26)(11,25)(12,27)(13,28)(15,18)(16,30)(17,31)(23,32)(24,29);(2,10)(5,6)(8,11)(9,26)(12,13)(15,17)(18,31)(19,21)(22,25)(23,24)(27,28)(29,32);(1,3)(2,11)(4,14)(5,15)(6,17)(7,16)(8,10)(9,25)(12,23)(13,24)(18,19)(20,30)(21,31)(22,26)(27,32)(28,29)Polycyclic presentation:
2^9|f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,f^a=fi,h^a=hi,c^b=ch,d^b=dfh,f^b=fi,g^b=gi,h^b=hi,h^c=hi,f^d=fi,h^d=hi,g^e=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72338591 |
| maximal normal subgroup of | #72338591 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6539304);