Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #494749 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1898 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #60367 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,9,7,10,5,12)(3,13,17,29,18,30,16,31)(4,19,14,27,21,11,22,24)(8,23,25,20,26,32,28,15);(1,3,14,23)(2,8,24,29)(4,20,6,16)(5,17,21,15)(7,18,22,32)(9,25,11,13)(10,26,27,31)(12,28,19,30);(2,9)(4,21)(5,6)(8,25)(10,12)(11,24)(13,29)(15,20)(16,17)(19,27)(26,28)(30,31);(2,10)(8,26)(9,12)(11,19)(13,30)(24,27)(25,28)(29,31)Polycyclic presentation:
2^9|a^2=g,b^2=eg,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=eh,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72231746, #72237542, #72237544, … |
| maximal normal subgroup of | #72231746, #72237542, #72237544, … |
| maximal quotient of | C2×#494758, #72280625, #72280695, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401954);