Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498898 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2021 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,10)(5,8)(6,9)(7,13)(11,18)(14,25)(15,26)(16,22)(17,23)(19,32)(20,28)(21,29)(24,31)(27,30);(1,3)(2,7)(4,13)(5,14)(6,15)(8,21)(9,20)(10,12)(11,19)(16,29)(17,28)(18,30)(22,25)(23,26)(24,27)(31,32);(2,8)(5,18)(7,19)(9,11)(10,22)(12,25)(13,28)(14,30)(16,31)(20,21)(23,24)(26,27);(2,9)(7,20)(8,11)(10,23)(12,26)(19,21)(22,24)(25,27)Polycyclic presentation:
2^9|f^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,g^a=ghi,c^b=cg,h^b=hi,e^c=eh,f^c=fi,g^e=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72210522, #72231746, #72236900, … |
| maximal normal subgroup of | #72210522, #72231746, #72236900, … |
| maximal quotient of | C2×#498898, #72280814, #72280866, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406094);