Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500919 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2669 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,11,7,13,26,12)(3,14,18,27,17,30,19,8)(4,20,22,5,10,25,23,9)(15,16,29,24,31,32,21,28);(1,3)(2,8)(4,21)(5,24)(6,19)(7,17)(9,28)(10,29)(11,30)(12,14)(13,27)(15,23)(16,25)(18,26)(20,32)(22,31);(1,4)(2,9)(3,15)(5,13)(6,22)(7,10)(8,16)(11,20)(12,25)(14,24)(17,31)(18,29)(19,21)(23,26)(27,32)(28,30);(1,5)(2,10)(3,16)(4,13)(6,20)(7,9)(8,15)(11,22)(12,23)(14,21)(17,32)(18,28)(19,24)(25,26)(27,31)(29,30);(3,17)(8,27)(14,30)(15,31)(16,32)(18,19)(21,29)(24,28)Polycyclic presentation:
2^9|a^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^b=ei,f^b=fi,h^b=hi,g^c=gi,h^c=hi,f^d=fi,g^d=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72213913, #72214082, #72215721, … |
| maximal normal subgroup of | #72213913, #72214082, #72215721, … |
| maximal quotient of | #72250753, C2×#7500919, #72252198, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408115);