Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7266801 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #46996 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
1695 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)(4,8)(5,9)(6,10)(7,14)(12,19)(13,24)(15,22)(16,23)(17,20)(18,26)(21,28)(25,30)(27,31)(29,32);(1,3)(2,7,10,11)(4,12)(5,17,18,13)(6,14)(8,21,23,19)(9,24)(15,29,30,27)(16,28)(20,26)(22,31)(25,32);(1,4)(2,8)(3,12)(5,15)(6,16)(7,19)(9,25)(10,23)(11,21)(13,27)(14,28)(17,29)(18,30)(20,32)(22,26)(24,31);(1,5)(2,9)(3,13)(4,15)(6,18)(7,20)(8,22)(10,26)(11,24)(12,27)(14,17)(16,30)(19,31)(21,32)(23,25)(28,29);(4,16)(7,11)(8,23)(12,28)(15,30)(20,24)(22,25)(27,29)Polycyclic presentation:
2^9|b^2=fgh,b^a=bf,c^a=cg,e^a=ef,h^a=hi,c^b=ch,d^b=df,e^b=eh,f^b=fi,g^b=gi,d^c=dh,e^c=ei,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237742, #72237743, #72237815, … |
| maximal normal subgroup of | #72237742, #72237743, #72237815, … |
| maximal quotient of | C2×#7266801, #72283460, #72283479, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7173997);